Name:
Skill Sheet 1.1
Solving Equations
Concepts in physics can often be expressed using formulas written as mathematical equations. To make
calculations using these formulas, the equations must be rearranged and solved for the variables that are used
in the relationship. As you solve equations, your goal is to isolate the unknown variable on one side of the
equation. As you proceed, you must use transformations that produce equations that are equivalent to the
original. Several inverse operations can be used.
1. Simple examples of solving equations
Sample
To solve it . . .
Solution
Addition:
n – 5 = 12
Add 5 to each side.
n = 17
Subtraction:
q + 4 = 16
Subtract 4 from each side.
q = 12
Multiplication:
t/
Multiply both sides by 2.
t = 36
Division:
4r = 24
Divide both sides by 4.
r=6
Multiplying by a reciprocal:
12 = 3/4 y
Multiply both sides by 4/3.
y = 16
2
= 18
2. Solving equations that require transformations
Solving some equations may require two or more transformations. To do this, follow the steps below:
•
Simplify one or both sides of the equation, if necessary.
•
Use inverse operations to isolate the variable.
Sample
To solve it . . .
1/
3a
+ 8 = 24
Write the original equation.
1/
3a
+ (8 – 8) = 24 – 8
Subtract 8 from both sides.
1/
3a
= 16
Simplify.
3(1/3)a = 3(16)
Multiply both sides by 3.
a = 48
1
Skill Sheet 1.1 Solving Equations
3. Solving equations that have variables on both sides
Some equations have variables on both sides of the equal sign. To solve these equations, collect the variable
terms on the side with the greater variable coefficient.
Sample
To solve it . . .
1/
Write the original equation.
4 (12x
+ 16) = 10 - 3(x – 2)
4(1/4)(12x + 16) = 4[10 - 3(x-2)]
Multiply both sides by 4.
12x + 16 = 40 – 12x + 24
Simplify.
12x + 16 – 16 = 64 – 16 – 12x
Subtract 16 from both sides.
12x + 12x = 48 – 12x + 12x
Add 12x to both sides.
24x = 48
Divide each side by 24.
x=2
4. Solving equations that contain radicals
Some equations contain radicals. Squaring both sides of an equation often simplifies the solution. When squaring
both sides of an equation, often an answer is introduced that is not a solution. These are referred to as extraneous
solutions. You should always substitute your answers into the original equation to check for extraneous solutions.
Sample
To solve it . . .
x–7 = 0
Write the original equation.
x–7+7 = 0+7
Add 7 to both sides.
2
( x) = 7
2
Square both sides.
x = 49
Sample
To solve it . . .
Write the original equation.
x + 13 = 0
x + 13 – 13 = 0 – 13 Subtract 13 from both sides.
2
( x ) = ( –13 )
x = 169
169 + 13 ≠ 0
2
Square both sides.
Simplify.
Checking the solutions in the original equation, you see that x = –1 is not a valid
solution.
2
Skill Sheet 1.1 Solving Equations
Sample
To solve it . . .
(x + 2) = x
2
( x + 2) = x
x+2 = x
Write the original equation.
2
2
2
Square both sides.
Simplify.
0 = x –x–2
Write in standard form.
0 = ( x – 2 )( x + 1 )
Factor.
x = 2 and x = -1
Checking by substitution, you see that the equation has no solution.
(2 + 2) = 2
( – 1 + 2 ) ≠ –1
Checking the solutions in the original equation, you see that x = 2 is a valid solution, but
x = -1 is not a valid solution.
5. Solving equations with quadratic equations
Some common relationships in physics are expressed using quadratic equations. The general
expression for a quadratic equation is: ax2 + bx + c = 0. To find the solutions for these
equations it is often necessary to use the quadratic formula. The formula is:
2
– b ± b – 4ac
x = -------------------------------------2a
Use of this equation may be time-consuming, but it gives an exact solution.
Example: A path of a certain waterfall as it tumbles over a vertical cliff can be given by the equation:
2
h = – 6.03 x + 901
where h = the height of the waterfall and x is the horizontal distance (in feet) from the base of the cliff to where
the water contacts the ground. How far from the base of the cliff does the water contact the ground?
Since h = 0 when the water strikes, the quadratic equation for this problem is written: 0 = -6.03x2 + 901
2
b ± b – 4ac , the solution is written as:
Using the quadratic formula, x = –-------------------------------------2a
– 0 ± 0 + ( – 4 ) ( – 6.03 )901x = ----------------------------------------------------------------2 ( – 6.03 )
x ≈ 12.3 or –12.3
The distance from the base of the cliff to where the water makes contact with the ground is 12.3 feet. The answer,
x = –12.3, is a meaningless value since the distance is in front of the cliff and therefore a positive number.
In addition to the techniques listed above, both quadratic equations and linear equations may be solved and
checked by graphing.
3
Skill Sheet 1.1 Solving Equations
6. Problems
For each of the following problems provide a solution by:
•
Writing an equation or equations that can be used to determine the unknown.
•
Solving the equation(s) for the unknown
1.
Jackie Joyner-Kersee won the gold medal in the Olympic heptathlon in both 1988 and 1992. Her score in
1992 was 7,044 points. This was 247 points higher than her score in 1988. What was her 1988 score?
2.
You decide to ride an elevator until it takes you to the ground floor without pushing any buttons. The
elevator takes you up 4 floors, down 6 floors, up 1 floor, down 8 floors, down 3 floors, up 1 floor, and then
down 6 floors to ground level. On what floor did you begin? (HINT: Note that the first floor is “1” not “0.”)
3.
A peregrine falcon can dive at speeds of 130 kilometers per hour. If at a height of 1.5 kilometers above the
ground the falcon spotted a grouse sitting in the top of a tree 12 meters tall, what is the least amount of time,
measured in seconds, it would take the falcon to reach the grouse?
4.
The distance a fire hose will project its water is given by the formula: d = n/2 + 26 where d is the maximum
distance in feet the hose will spray and n is the nozzle pressure measured in pounds per square inch. How
much pressure would be required to reach a flame 100 feet from the hose?
5.
Tim has fallen through the ice while ice skating. If his body temperature reaches 35° Celsius, he will
experience hypothermia, a possibly life-threatening condition. An EMT records his temperature as
94° Fahrenheit (°F = 9/5°C + 32). Will he experience hypothermia?
6.
Beth and Bob live 100 kilometers apart. They decide to ride their bicycles to meet one another. Beth rides
west toward Bob at 23 kilometers per hour. Bob starts at the same time and rides east toward Beth at
14 kilometers per hour. How long does it take for them to meet?
4
Skill Sheet 1.1 Solving Equations
7.
The period of a mass suspended on a vibrating spring can be approximated by the equation:
T = 2π m
---k
where T is the period, m is the mass of the object, and k is the spring constant of the spring. What is the
spring constant of a spring whose period of vibration is 0.1922 seconds when a mass of 0.010 kilogram is
suspended from the spring?
8.
The path of a diver diving from a 10-foot high diving board is given by the equation:
2
h = – 0.44x + 2.61x + 10
where h is the height of the diver above the water and x is the horizontal distance of the diver from the end of
the diving board. Calculate the horizontal distance from the end of the board at which the diver enters the
water. Assume that the height of the diver above the water is zero.
5
Name:
Skill Sheet 1.2
Galileo Galilei
Galileo Galilei was a mathematician, scientist, inventor, and astronomer. His observations led to significant
advances in our understanding of pendulum motion and free fall. He invented a thermometer, water pump,
military compass, and microscope. He refined a Dutch invention, the telescope, and used it to revolutionize
our understanding of the solar system.
Galileo Galilei was born in Pisa, Italy, in 1564. His father, a musician and wool trader,
hoped his son would find a more profitable career. He sent Galileo to a monastery
school at age 11 to prepare for medical school. After four years, Galileo had decided to
become a monk. His family had daughters who would need dowries in order to marry,
and his father planned on Galileo’s financial help. Galileo was hastily withdrawn from
the monastery school.
Two years later, he enrolled as a medical student at the University of Pisa, though his
real interest was in mathematics and natural philosophy. It soon became apparent that
Galileo did not intend to apply himself to medical studies and it was finally agreed that
he could study mathematics instead.
Galileo was insatiably curious. At age 20, he watched a lamp swinging from a cathedral ceiling. He used his
pulse as a makeshift stopwatch and discovered that the lamp’s long and short swings took the same amount of
time. He wrote about this in an early paper titled “On Motion.” Years later, he drew up plans for a new invention,
a pendulum clock based on his discovery.
Galileo began teaching at the University of Padua in 1592, and stayed for 18 years. Here he invented a simple
thermometer, a water pump, and a compass for accurately aiming cannonballs. He also performed experiments
with falling objects, using an inclined plane to slow the object’s motion so it could be more accurately timed.
Through these experiments, he realized that all objects fall at the same rate unless acted on by another force.
In 1609, Galileo heard that a Dutch eyeglass maker had invented an instrument that made things appear larger.
Soon he had crafted his own ten-powered telescope. The Senate in Venice was impressed with its potential
military uses and Galileo’s finances improved. In a year, he had refined his invention to a 30-powered telescope.
Galileo’s curiosity now turned toward the skies. Using his telescope, he discovered craters on the moon,
sunspots, Jupiter’s four largest moons, and the phases of Venus. His observations led him to conclude that Earth
could not possibly be the center of the universe, as had been commonly accepted since the time of the GrecoEgyptian astronomer Ptolemy in the second century. Instead, Galileo was convinced that fifteenth-century
mathematician Nicolaus Copernicus must have been right: The sun is in the center of the universe and the planets
revolve around it.
Galileo published his conclusion in February 1632. He chose to present his ideas in the form of a conversation
between two characters, the one representing Ptolemy’s view seeming foolish and bullheaded. This was
provocative because the Roman Catholic Church subscribed to Ptolemy’s ideas. That same year, Galileo was
brought before the Inquisition in Rome and convicted of heresy for promoting Copernicus’ ideas. He was
sentenced to house arrest, and lived until his death in 1642 watched over by Inquisition guards.
Questions
1. In your opinion, which of Galileo’s ideas or inventions had the biggest impact on history? Why?
2. Research one of Galileo’s inventions and draw a diagram showing how it worked.
1
Name:
Skill Sheet 2.1
International System of Units
In ancient times, as trade developed between cities and nations, units of measurement were developed to
measure the size of purchases and transactions. Greeks and Egyptians based their measurements of length on
the human foot. Usually, it was based on the king’s foot size. The volume of baskets was measured by how
much goatskin they could hold. Can you see how this could lead to disputes among merchants and their
customers? The International System of Units resolves this problem and others by providing a standard,
interrelated, and reproducible system of measurement.
1. A short history of measurement
The eighteenth century was a time of great beginnings in science. However, by century’s end, scientists found
that their system of measures was increasingly burdensome. Measurements such as the foot were not well
standardized and made it hard to communicate observations. A system that allowed scientists to reproduce and
verify each other’s data was needed.
The metric system was developed to fulfill this need. The system’s basic unit
for measuring length was called the meter, after the Greek word metron
meaning “measure.” But the metric system was not created in a single
development. For example, there were two ideas for the meter—one that used
the length of a pendulum and another that used a fraction of the distance
between Earth’s equator and the north pole. The north pole-to-equator line was
chosen in one of a series of decisions that shaped the metric system. Today, the
General Conference on Weights and Measures, or CGPM (Conférence
Générale des Poids et Mesures), has responsibility for these decisions.
The United States began to incorporate the use of the metric system the late
1800’s. However, most Americans still use the US Customary System
(inherited from the British Imperial System) of feet, inches, and pounds. Not only scientists but most countries—
even England—use what was named the International System of Units. In all these countries, your car’s speed is
measured in kilometers per hour, its gasoline in liters, the cheese you buy in grams, and the temperature in
degrees Celsius.
2. Today's International System of Units
The 11th General Conference on Weights and Measures in 1960 made some of the most important recent
revisions to the universal measurement system. A meter was defined as the distance light travels in a small
fraction of a second. A kilogram was reaffirmed as the mass of platinum-iridium kept in Paris. The International
System of Units was renamed Système International d’Unités and the new “modernized” metric system given the
official symbol SI.
Most students regard “metrics” as a set of memorized prefixes that increase a measurement by tens. This is
certainly true, but it overlooks one of the most important characteristics of SI units. The SI unit for volume, the
liter, is derived from the meter. A liter is that volume contained in a cube that measures 10 centimeters on each
side. The SI unit for weight, the kilogram, was originally the weight of one liter of pure water at standard
temperature and pressure. Although today a kilogram is defined by the prototype platinum-iridium kilogram kept
in Paris, both definitions are close enough to be interchangeable except in the most precise work. This
interrelated characteristic makes SI measurements very easy for scientists and non-scientists.
1
Skill Sheet 2.1 International System of Units
Consider the carpenter who is installing a hot tub. He needs to know the weight of the tub filled with water to
determine whether to strengthen the supports:
Using English units
Using SI units
The carpenter measures a tub interior and finds that
it is 6 feet long, 2 feet deep, and 3 feet wide. He
calculates the tub’s volume to be 36 cubic feet.
The carpenter measures the interior of a tub and
finds that it is 2 meters long, 60 centimeters (0.6 m)
deep, and 1 meter wide. He calculates the tub’s
volume to be 1.2 cubic meters.
There are 7.48 gallons per cubic foot. Therefore, he
multiplies the volume by 7.48 gallons/cubic foot
and finds that the tub holds 269.3 gallons of water
when filled.
He knows that a cubic meter is equivalent to 1,000
liters, so he shifts the decimal and the volume
becomes 1,200 liters.
He multiplies gallons by 8.36 pounds/gallon and
finds the water weight will be 2251.3 pounds.
He also knows that a liter of water weighs
1 kilogram, so in shifting the decimal he arrived at
the weight directly–1,200 kilograms.
He did all of the above in his head.
3. SI prefixes
Prefixes in the SI system indicate the multiplication factor to be used with the measurement unit. For example,
the prefix kilo multiplies the unit by 1,000. A kilometer is equal to 1,000 meters. A kilogram equals 1,000 grams.
Prefix
Symbol
Multiplication factor
pico–
p
0.000000000001
= 10-12
nano–
n
0.000000001
= 10-9
micro–
µ
0.000001
= 10-6
milli–
m
0.001
= 10-3
centi–
c
0.01
= 10-2
deci–
d
0.1
= 10-1
1
= 100
No prefix
deka–
da
10
= 101
hecto–
h
100
= 102
kilo–
k
1,000
= 103
mega–
M
1,000,000
= 106
giga–
G
1,000,000,000
= 109
tera–
T
1, 000, 000, 000, 000
= 1012
2
Skill Sheet 2.1 International System of Units
4. Practical units of SI measurement
In practice, many of the possible prefix and unit measurements are seldom used. In the table below, the most
commonly used SI units of length, volume, and weight are provided.
The most commonly used SI units
Prefix
Length
Volume
Weight
milli-
millimeter
milliliter
milligram
centi-
centimeter
meter
liter
gram
kilometer
(meter3)
kilogram
deci(unit)
decahectokilo-
Consider this progression:
millimeter −−>
centimeter −−>
meter −−>
kilometer −−>
Now the prefix multipliers:
0.001
0.01
1
1,000
Finally, consider the magnitude of change between each of these steps:
millimeter to centimeter = 10
centimeter to meter
= 100
meter to kilometer
= 1,000
This progression explains why there are gaps in the table and illustrates the practical side of measurement. As the
quantity to be measured increases, the size of the most practical unit increases geometrically. If a meter is too
small for a measurement, the next largest prefix-unit, hectometers, will probably not be large enough either.
Here is another example of SI units in practice. The prefix kilo can be joined with the unit liter to form kiloliter,
but you will never see it written and here’s why. Small laboratory ware and kitchenware are calibrated in
milliliters and liters. Remember that a liter of water weighs one kilogram (2.2 pounds), which is easy enough to
pick up. But a kiloliter is 1,000 times heavier! How would you measure and handle that amount of water? Easy—
you would measure its container in meters as the carpenter did in the hot tub example above.
3
Skill Sheet 2.1 International System of Units
5. Comparing SI units
1.
2.
3.
Is a cubic meter equivalent to another measure?
Large volumes are determined by measuring the length, width, and height of their containers. The result is
expressed as the cube of the unit of length used. Because large containers are measured in meters, the result
is expressed in cubic meters. Can you determine another SI unit that is equal to a cubic meter?
•
Remember that a liter is a cube that measures 10 centimeters on each side.
•
Now visualize a cubic meter. How many liter-cubes would line one side? (10 liter-cubes)
•
How many liter-cubes would fit into your virtual cubic meter? (1,000 liter-cubes)
•
How many liters are in a kiloliter? Use the prefix table to answer this question. (1,000 liters)
•
What is the relationship between a cubic meter and a kiloliter? (They are equivalent.)
Now you realize that in the practical world of measurement, large volumes are always measured in cubic
meters. That is why the table in Part 4 shows meters3 in parentheses where kiloliters would have appeared.
Lake volumes are measured from maps in cubic meters, natural gas is delivered to homes in cubic meters,
and topsoil lost to erosion is measured in cubic meters
What is the relationship between a cubic centimeter and a milliliter?
•
What is the length of one side of a liter-cube in centimeters? Use the information found in Practical units
of SI measurement, above. (10 cm)
•
What is the volume of a liter in cubic centimeters? (Use the volume formula, l × w × h. (1,000 cm3)
•
How many milliliters are in a liter? Use the prefix table to answer this question. (1,000 mL)
•
What is the relationship between a cubic centimeter and a milliliter? (They are equivalent.)
What is the relationship between a milliliter of water and a gram?
A milliliter of salt water weights more than a milliliter of fresh water. Therefore, to discover the relationship
between a milliliter of water and a gram, we must define the nature of the water. We will use pure water at
standard temperature and pressure.
•
How much does a liter of pure water at standard temperature and pressure weigh? (1 kg; see Part 2)
•
How many grams are in a kilogram? Use the prefix table to answer this question. (1,000 g)
•
How many milliliters are in a liter? Use the prefix table to answer this question. (1,000 mL)
•
What is the relationship between a milliliter of pure water at standard temperature and pressure and a
gram? (One milliliter weighs one gram.)
It’s important to note that the results of the first two SI challenges were equivalent; you can exchange one for the
other. Although the result of this challenge is numerically equivalent, volume and weight are completely different
concepts. That is why you must state that a milliliter of water WEIGHS one gram, not that a milliliter of water
equals one gram.
4
Skill Sheet 2.1 International System of Units
6. Applying the results of the SI challenges
European kitchens always include a tare scale. A tare scale weighs just like any other scale except that it has a
button that returns the weight to zero even if there is something on it. Cooking with a tare scale and SI units is
easy. Here is how it’s done:
Many recipes like brownies and flavored noodles require an amount of water and milk to be added to a mix. How
might a tare scale allow you to prepare flavored noodles using only a tare balance, a pot, and a mixing spoon?
•
Place the pot with the dry noodles on the balance. Press tare to zero the scale.
•
Run water into the pot until the balance reads in grams the amount of water needed in milliliters.
•
Press tare again to zero the balance. Pour milk from the container until the scale reads in grams the amount
of milk needed in milliliters. Then, heat and stir the noodle, water, and milk mixture. Bon appétit!
7. Converting between two SI Units
Here is how you convert one SI unit to another.
1.
2.
3.
4.
5.
Ask yourself whether the new unit is larger or smaller than the old unit. For example, in converting from
meters to centimeters, the new unit is smaller than the old unit.
If the new unit is smaller, the new quantity must be larger to maintain equality. If the new unit is larger, the
new quantity must be smaller. To make this idea clear, think about a mother and a small child walking across
the room. The mother takes a few large steps, but the child must take many small steps.
If the quantity of the new unit must be larger, the decimal is moved to the right. If the quantity of the new
unit must be smaller, the decimal is moved to the left.
Find the multiplication factor for the old and new units in the SI prefixes table in Part 3. Subtract the
exponents algebraically. Disregard the sign of the result.
Move the decimal in the old quantity in the direction found in Step 2. Move the number of places found in
Step 4. Add zeros if necessary. Write this number as the new quantity.
Example:
How many centimeters equal 2.35 meters?
1.
2.
3.
4.
The new unit (centimeters) is smaller than meters. Therefore, the new quantity must be larger.
The decimal must move right.
The multiplication factor exponent for meter = 0, multiplication factor exponent for centimeter = -2.
Difference without regarding sign = 2.
Therefore, the decimal moves right two places. The new quantity = 235 centimeters.
Practice:
1.
2.
3.
4.
5.
6.
3.45 milligrams = _______________ grams
3.004 meters = _______________ centimeters
112.3 grams = _______________ kilograms
6567.09 millimeters = _______________ centimeters
5.2 liters = _______________ milliliters
How many centimeters are in a kilometer? _______________ centimeters
5
Name:
Skill Sheet 2.2
Converting Units
All measurements have two parts, an amount shown as a number and a unit shown as a word. For example,
your height might be 64 (the amount) inches (the unit). This measurement is equivalent to 5 feet 4 inches.
Why might you use feet and inches to describe your height versus inches only? The type of unit you use
depends on how large or how small a measurement is. For example, the distance to your school might be
158,400 inches or 2.5 miles. Do you see why you might use miles to describe this distance? You will practice
converting between units in this skill sheet.
1. Canceling units or ‘crossing out’
Canceling units is the key to converting units. Here are the four concepts involved. Take it a step at a time to see
what is happening.
1.
One factor multiplied by a fraction is equal to a single fraction:
1m
cm × 1 m23 cm × ------------------ = 23
-----------------------------100 cm
100 cm
2.
Units are just like algebraic variables. Think of them as multiplied by their amounts:
23
cm × 1 m- 23 × cm × 1 × m
-----------------------------= ----------------------------------------100 cm
100 × cm
3.
Anything divided by itself is equal to 1:
1--- = 1
1
4.
23
------ = 1
23
cm
------- = 1
cm
One times anything is equal to that thing, so multiplying by 1 does not change the value:
1×3 = 3
1 × cup = cup
1 × gram = gram
See how each of these concepts works together. The first two concepts are applied to three separate fractions:
1,000 m 1 mile
100 × km × 1,000 × m × 1 × mile100 km × -------------------- × -------------------- = ------------------------------------------------------------------------------1 km
1,609 m
1 × km × 1,609 × m
Scanning this combined fraction above, we see two cases of like terms in the numerator and the denominator.
Kilometers (km) and meters (m) appear in the numerator and the denominator. The third concept above says that
anything divided by itself is equal to 1. Therefore, we can cancel out these terms.
1
1
1,000 m 1 mile
100
×
km
×
1,000
×
m
× 1 × mile
100 km × -------------------- × -------------------- = -------------------------------------------------------------------------------1 km
1,609 m
1 × km × 1,609 × m
The fourth concept says that the two 1’s that resulted from canceling will not change the final value. Note that the
single remaining unit (miles) is carried over as the unit in the result. To provide a final check, determine whether
the resulting unit—in this case, miles—makes sense. Does it make sense to say that 100 kilometers is equal to
62.1 miles?
100 × 1 × 1,000 × 1 × 1 × mile- = 62.1 miles
------------------------------------------------------------------------1 × 1 × 1,609 × 1
1
Skill Sheet 2.2 Converting Units
2. Choosing conversion factors
The first step of converting units is to select a conversion factor that matches the units of the problem. Sometimes
you need more than one conversion factor to solve the problem. For each problem below, circle one or more
conversion factors that you would use to solve each problem. You do not need to solve these problems.
Problem
Conversion Factors
Example:
3, 043 meters equals how many kilometers?
10
mm---------------1 cm
10
cm-------------1m
1 km -------------------1, 000 m
The problem involves meters and kilometers. The circled conversion factor shows the relationship between
meters and kilometers (1 km = 1,000 m) so this is the conversion factor to use.
1.
183 cm equals how many meters?
10
mm---------------1 cm
1m----------------100 cm
1,000
m------------------1 km
1.
53 mm equals how many centimeters?
10
mm---------------1 cm
10
cm-------------1m
1,000
m------------------1 km
1.
73,680 cm equals how many kilometers?
10
mm---------------1 cm
1m----------------100 cm
1 km ------------------1,000 m
3. Applying the conversion factor correctly
Conversion problems are solved with one or more conversion factors. In the conversion process, the starting unit
is canceled, leaving only the ending unit in the answer. To do this, the unit to be canceled must appear in the
denominator of the conversion factor. Sometimes this requires you to invert the conversion factor. Solve each
problem with the conversion factor as is or inverted.
Problem
Conversion factors
Example:
1.5 miles is equal to how many kilometers?.
1 kilometer
1.5 miles × ---------------------------- = 2.4 kilometers
0.624 miles
1.
0.624
miles--------------------------1 kilometer
1 kilometer--------------------------0.624 miles
In this problem, the conversion factor
needs to be inverted.
4.3 centimeters is equal to how many millimeters?
10
millimeters---------------------------------1 centimeter
M
1.
8,700 milligrams is equal to how many grams?
1.
4.3 Astronomical Units is equal to how many kilometers?
10
milligrams
--------------------------------1 gram
2
149,597,870.7
kilometers-----------------------------------------------------------1 Astronomical Unit
Skill Sheet 2.2 Converting Units
4. Practice problems
Solve the following problems using the conversion factors found on the back cover of your text, Foundations of
Physics. Additional conversion factors are included with each problem. Round final answers to the nearest tenth.
In problems with more than one conversion factor, work each conversion factor one at a time, working from left
to right. Treat the result of the first conversion as though it were the beginning of a new problem and ensure that
the new unit to be canceled is in the denominator of the next conversion factor.
Use the following rules to check your work:
•
If the ending unit is larger than the starting unit, the ending amount must be
smaller that the starting amount. Example: 12 eggs = 1 dozen
Dozen is a larger unit than egg; 1 is smaller than 12.
•
If the ending unit is smaller, the ending amount must be larger than the
starting amount.
Example: 1 meter = 100 centimeters
Centimeter is smaller than meter; 100 is larger than 1.
•
The final unit after conversion must answer the original question.
1.
Fill in the following table:
Starting amount and unit
Ending amount and unit
3.0 inches
_____ meters
3.7 gallons
_____ liters
47.0 pounds
_____ kilograms
3.0 pints
_____ liters
230 grams
_____ kilograms
42 millimeters
_____ centimeters
1,000 milliliters
_____ liters
24.3 meters
_____ kilometers
Conversion factors: 0.4536 kilograms = 1 pound or
2.
0.4536
kg----------------------;
1 pound
8 pints = 1 gallon or
8 pints-----------------1 gallon
The volume of a European hot tub is 2,800 liters, but the building code for floor joist size to support the tub
is in gallons. How many gallons should the builder use to calculate the weight of the filled tub?
3
Skill Sheet 2.2 Converting Units
3.
A bullet fired from a .22-caliber rifle leaves the barrel at 1,200 feet per second. How fast is that in meters per
second?
meters--------------------------------Conversion factor: 0.3048 meters = 1 foot or 0.3048
1 foot
4.
One reason that SI units are not popular in the United States is that converting English units directly into SI
units results in numbers with decimals. What would the weight be of a 2-pound can of coffee in grams?
Conversion factor:
grams
----------------------------453.6 grams = 1 pound or 453.6
1 pound
5.
The beverage industry in the United States has been eager to use SI units. One liter of a beverage has 1,000
milliliters. Calculate how many milliliters are in one quart. Why do you think it would be a good marketing
move to sell beverages by the liter rather than by the quart?
Conversion factor:
4 quarts
4 quarts = 1 gallon or -----------------1 gallon
6.
A young French girl went to the market and bought 200 grams of cheese for her mother. About how many
ounces of cheese did she buy?
Conversion factors:
grams
----------------------------453.6 grams = 1 pound or 453.6
1 pound
16 ounces = 1 pound or
7.
16 ounces
-----------------------1 pound
Challenge problem: Here is a good multipart problem that gives you an eye-opening idea of the immense
distances of space. It is also completely imaginary. Although we know sound cannot travel through a
vacuum, imagine that sound can travel in space at the same speed it travels through air under standard
conditions. Our sun has just erupted in an enormous solar flare. We will see it in about 8 1/2 minutes because
of the speed of light. But how long after the flare would we hear it under these imaginary conditions? Round
to the nearest whole number after each step.
Conversion factors:
Distance to sun = 93,000,000 miles
Speed of sound under standard conditions = 343 meters per second.
One kilometer = 0.62 miles
One kilometer = 1,000 meters
One hour = 3,600 seconds.
One year = 8,766 hours.
HINT: Convert the sun's distance to meters and calculate the number of seconds, then convert the number of
seconds to years.
4
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Skill Sheet 2.3
Scientific Notation
Visualizing and working with very large and very small numbers is easier when you use scientific notation. In
this skill sheet, you will practice using scientific notation.
1. When is scientific notation useful?
Scientific notation is useful for large and small numbers, but not when the numbers need to be highly precise.
This is because at high precision, each number must be displayed. The two examples below illustrate when
scientific notation is useful and when it is not.
Scientific notation is useful when the number is not very precise:
Speed of light = 300,000,000 cm/sec = 3.00 × 108 cm/sec
Here, the speed of light is not a precise number. We do not know if the speed of
light is 299, 999, 999 cm/sec or 300,000,000.321 cm/sec.
Scientific notation is not useful when the number is precise:
A precise length of 1 meter = 1,650,763.73 wavelengths of orange-red light
from a krypton-86 lamp.
We cannot easily write this number using scientific notation.
A
B
1. Which of the following numbers in these
sentences would be useful to write in
scientific notation?
The amount of money he had
in his bank account was
$30,892.23.
The net worth of that
businessman is
$53,000,000.000.
2. Which of the following numbers should
not be written in scientific notation?
503, 099, 111, 834.45
0.00000000672
2. Why is scientific notation useful?
Multiplying and dividing in scientific notation is so easy that you can often make estimates in your head. To
multiply in scientific notation, multiply the two bases and add the two exponents. For example, you can easily
calculate the square of the speed of light (3.00 × 108 cm/sec) using scientific notation:
Multiply bases:
Add exponents:
Assemble the number:
3.00 × 3.00 = 9.00
108 × 108 = 1016
9.00 × 1016 cm/sec
Dividing two numbers in scientific notation is similar to multiplying them. To divide, the first base is divided by
the second base. Then the second exponent is subtracted from the first.
4.00 × 108 ÷ 2.00 × 102
=
Divide bases:
Subtract exponents:
Assemble the number:
4.00 ÷ 2.00 = 2.00
108 ÷ 102 = 10(8 - 2) = 106
2.00 × 106
1
Skill Sheet 2.3 Scientific Notation
3. Converting from standard notation to scientific notation
•
Note the position of the decimal in the standard notation number. If there is no decimal, add one at the end of
the number.
•
Count the number of places as you move the decimal until the base number is equal or greater than 1, but
less than 10.
•
Usually the base is shown to two decimal places. Round the base number to two decimal places if necessary.
•
Write the exponent, equal to the number of places moved, as a power of ten. If you moved the decimal to the
left, the exponent is positive. If you moved it to the right, the exponent is negative.
Convert these standard notation numbers to scientific notation:
Standard notation
1.
6,700,000
2.
300,000,000
3.
3,600,000
4.
0.000645
5.
150,000,000
6.
0.001
7.
186,000
8.
0.001341
Scientific notation
4. Converting from scientific notation to standard notation
•
If the exponent is negative, count places left from the decimal in the base number equal to the exponent.
•
If the exponent is positive, count places right from the decimal in the base number equal to the exponent.
4.00 × 108 = 400 × 106 Now add six more zeros to 400 to make 400,000,000
Convert these scientific notation numbers to standard notation:
Scientific notation
1.
3.47 × 108
2.
7.94 × 103
3.
1.96 × 10-3
4.
4.50 × 109
5.
9.61 × 10-6
6.
2.02 × 105
7.
7.01 × 10-10
8.
4.44 × 101
Standard notation
2
Skill Sheet 2.3 Scientific Notation
5. Scientific notation using a calculator
You can convert between standard and scientific notation with a scientific calculator. Calculators vary in
keystrokes used, but most are similar to the keystroke steps listed here. Compare these steps to the instructions
for your scientific calculator.
To enter a number in scientific notation:
•
Type the base number using the number keys and the decimal key.
•
Press the exponent key. On some calculators this key is labeled <EE>. Other common exponent keys are
<Ex >, <Exp>, and <x10>. This automatically sets up the power of ten exponent entry.
•
If the exponent is negative, press the <(-)> and then type exponent. Doing this will make the exponent a
negative number. On some calculators, you type the negative sign after typing the exponent.
Check your work in Part 3 by using these steps with a calculator.
To convert a number in scientific notation to standard notation:
•
If the calculator has different display modes, set it for standard notation display.
•
Enter a number in scientific notation using the steps above.
•
Press the <=> key.
Check your work in Part 4 by using these steps with a calculator.
6. Scientific notation operations using a calculator
Calculators can do math operations (+, -, ·, ÷) in scientific notation and mixed standard and scientific notation.
The procedure is no different from what you have used with standard notation. Simply enter the first number, the
operator, the second number, and then type <=>. Some calculators allow the display mode to be changed between
standard and scientific notation. A calculator set to standard notation display mode allows you to divide two
numbers in scientific notation, but the result appears in standard notation. To display the result in scientific
notation, change the display mode. Solve these operations with your calculator in scientific notation mode.
1.
2.34 × 103 × 6 =
2.
2.34 × 103 × 1.10 × 104 =
3.
7.02 × 103 ÷ 2.34 × 102 =
4.
4.32 × 109 + 7.87 × 109 =
5.
5.21 × 10-4 – 3.01 × 10-4 =
6.
5.06 × 102 + 410 =
7.
7.22 × 105 × 3.33 × 103 =
8.
8.64 × 104 – 5.02 × 103 =
9.
9.87 × 10-7 ÷ 2.10 × 10-3 =
10.
6.88 × 10-3 + 2.45 × 10-3 =
3
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Skill Sheet 3.1
Speed Problems
This skill sheet will allow you to practice solving speed problems. To determine the speed of an object, you
need to know the distance traveled and the time taken to travel that distance. However, by rearranging the
formula for speed, v = d/t, you can also determine the distance traveled or the time it took for the object to
travel that distance, if you know the speed. For example,
Equation…
Gives you…
If you know…
v = d/t
speed
time and distance
d=v×t
distance
speed and time
t = d/v
time
distance and speed
1. Solving problems
Solve the following problems using the speed equation. The first problem is done for you.
1.
What is the speed of a cheetah that travels 112.0 meters in 4.0 seconds?
m- = 28
mspeed = d--- = 112.0
---------------------------t
4.0 sec
sec
2.
A bicyclist travels 60.0 kilometers in 3.5 hours. What is the cyclist’s average speed?
3.
What is the average speed of a car that traveled 300.0 miles in 5.5 hours?
4.
How much time would it take for the sound of thunder to travel 1,500 meters if sound travels at a speed of
330 m/sec?
5.
How much time would it take for an airplane to reach its destination if it traveled at an average speed of
790 kilometers/hour for a distance of 4,700 kilometers?
1
Skill Sheet 3.1 Speed Problems
6.
How far can a person run in 15 minutes if he or she runs at an average speed of 16 km/hr?
(HINT: Remember to convert minutes to hours)
7.
A snail can move approximately 0.30 meters per minute. How many meters can the snail cover in
15 minutes?
2. Unit conversion
So far we have been mostly using the metric system for our problems. Now we will convert to the English system
of measurement. Remember that there are 1,609 meters in one mile. Do not forget to include all units and cancel
appropriately. These questions refer to problems in Part 1.
1.
In problem 1.1, what is the cheetah’s speed in miles/hour?
28
m- -----------------1 mile - 3---------------------, 600 sec
----------×
×
sec 1,609 m
1 hour
=
63 miles
-------------------hour
2.
In problem 1.5, what is the airplane’s speed in miles/ hour?
3.
In problem 1.6, what is the runner’s distance traveled in miles?
4.
You know that there are 1,609 meters in a mile. The number of feet in a mile is 5,280. Use these equalities to
answer the following problems:
a. How many centimeters equals one inch?
b. What is the speed of the snail in problem 1.7 in inches per minute?
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Skill Sheet 3.2
Making Line Graphs
Graphs allow you to present data in a form that is easily and quickly understood. Graphs are especially good
for describing changing data. Here is how a line graph is made.
1. Examine your data
Graph data consist of data pairs. Each data pair represents two
variables. The independent variable will be plotted on the x(horizontal) axis and the dependent variable will be plotted on the
y-(vertical) axis. It’s much easier to find the dependent than the
independent variable. Test for the dependent variable by asking
yourself which one depends on the other. For example, in a data
set of money earned for hours worked, the dependent variable is
money earned. That is because the money depends on the hours
worked. Use this test to determine the dependent variable in any
data set.
Next, determine the numerical range between the smallest and
largest value for each variable. If you started working with zero
hours and worked 60 hours to earn this money, the range for hours
would be 60. During this time, if you started with $50 and finished
working with $320, the range for dollars would be 270. Be sure to
calculate the range for the independent and depend variables
separately.
2. Examine your graph
Check the space that you will use for your graph. If you are using a piece of graph paper, allow some space on the
side and bottom for labels and other information. Now count the number of lines right from the y-axis to nearly
the edge of the space. This is the maximum graph space that you have for the independent variable. Repeat this
process for the dependent variable by counting the number of lines up from the x-axis to nearly the top of the
space. This is the maximum graph space that you have for the dependent variable.
3. Determine the graph scale
You are now going to set the scale for the independent and dependent variables. It is important that you calculate
the scales separately. The independent and dependent variables usually have different scales. We know that the
dependent variable, money, has a range of $270. Now imagine that the graph space has a maximum of 20 lines on
the y-axis before it nearly runs off the page. The first line (the x-axis) will be labeled $50. What will the next
higher line be labeled? The point here is that if the value is too small, some of the money data will run out of the
graph space. But if the value is too large, the plotted line will be small and hard to read. Divide the number of
lines into the range to find a starting value. Increase the scale to an easier-to-use scale if necessary.
$270 ÷ 20 lines = $13.50/line
Increase to an easier scale: $15/line
1
Skill Sheet 3.2 Making Line Graphs
Now label the y-axis at $15 per line. It is not necessary to label each line; perhaps each fourth line as a multiple
of $60. Repeat this process for the independent variable.
1.
2.
Use the above information and the graphic to help you make a graph of the following data: (0, $50), (10,
$95), (20, $140), (30, $185), (40, $230), (50, $275), (60, $320). Use your own graph paper or the graph
paper on the last page of this skill sheet.
Use the data and the graph to determine the amount earned per hour during the 60 hours of work time.
4. Determine the independent and dependent variables
Two variables are listed in each row of the first two columns of the table below. Identify the independent and
dependent variable in each data pair. Rewrite the data pair under the correct heading in the next two columns of
the table. The first data pair is done for you.
Data pair not necessarily in order
Independent
Temperature
Hours of heating
Hours of heating
Reaction time
Alcohol consumed
Number of people in a family
Cost per week for groceries
Stream flow rate
Amount of rainfall
Tree age
Average height
5. Find the data range
Calculate the data range for each variable:
Lowest value
Highest value
0
28
10
87
0
4.2
-5
23
0
113
100
1250
2
Range
Dependent
Temperature
Skill Sheet 3.2 Making Line Graphs
6. Set the graph scale
Using the variable range and the number of lines, calculate the scale for an axis and then determine an easy-touse scale. Write the easy-to-use scale in the column with the heading “Adjusted scale.”
Range
Number of lines
13
24
83
43
31
35
4.2
33
12
33
900
15
Range ÷ Number of lines
3
Calculated scale
Adjusted scale
Skill Sheet 3.2 Making Line Graphs
Graph paper
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Skill Sheet 3.3A Analyzing Graphs of Motion Without Numbers
Graphs change columns of figures into images that are easy to interpret. Position-time and speed-time graphs
describe the movement of objects. Here are stories for you to tell as graphs and a graph for you to tell as a
story. Both will sharpen your graph interpreting skills.
1. Position-time graphs
Data
Remember the “The Three Little Pigs”?
•
The wolf started from his house.
•
Traveled to the straw house.
•
Stayed to blow it down and eat dinner.
•
Traveled to the stick house.
•
Again stayed, blew it down, and ate seconds.
•
Traveled to the brick house.
•
Died in the stew pot at the brick house.
The wolf started at his house, and the graph starts at the origin. Each time the wolf moves farther from his house,
the line moves upward with passing time. At each pig’s house, the line continues to the right but neither rises nor
falls, indicating that the wolf has stopped moving relative to his starting point. We can deduce that the pigs’
houses are generally in a line away from the wolf’s house and that the brick house was the farthest away. How
would the line look if the brick house were on the way back to the wolf’s house? Remember that position refers
to the starting point—in this case, the wolf’s house.
2. Speed-time graphs
A speed-time graph displays the speed of an object over time and is based on position-time data. You know that
speed is the relationship between distance and time, R = D/T. Look at the first part of the wolf’s trip. The line
rises steadily to a new position and a new time. It would be easy to calculate a speed for this first leg. What if the
wolf traveled this first leg faster? The new line would rise to the same position, but it would take less time. That
would make the new line steeper. Here is the speed-time graph for the wolf:
The wolf moved at the same speed toward his first two “visits.” His
third trip was slightly slower. Except for this slight difference, the wolf
was either at one speed or stopped. That is why this graph is so
angular.
1
Skill Sheet 3.3A Analyzing Graphs of Motion Without Numbers
3. Stories for you to tell as graphs
Read each of the following stories. Then sketch in the line for a position-time graph and a speed-time graph.
1.
2.
“Little Red Riding Hood.” Graph Red Riding Hood's movements.
Data:
•
Little Red Riding Hood set out for Grandmother’s cottage at a good walking pace.
•
She stopped briefly to talk to the wolf.
•
She walked a bit slower because they were talking as they walked to the wild flowers.
•
She stopped to pick flowers for quite a while.
•
Realizing she was late, Red Riding Hood ran the rest of the way to Grandmother’s cottage.
The Tortoise and the Hare. Use two lines to graph both the tortoise and the hare.
Data:
•
The tortoise and the hare began their race from the combined start-finish line.
•
Quickly outdistancing the tortoise, the hare ran off at a moderate speed.
•
The tortoise took off at a slow but steady speed.
•
The hare, with an enormous lead, stopped for a short nap.
•
With a start, the hare awoke and realized that he had been sleeping for a long time.
•
The hare raced off toward the finish at top speed.
•
Before the hare could catch up, the tortoise’s steady pace won the race with an hour to spare.
2
Skill Sheet 3.3A Analyzing Graphs of Motion Without Numbers
3.
The Skyrocket. Graph the altitude of the rocket.
Data:
•
The skyrocket was placed on the launcher.
•
As the rocket motor burned, the rocket flew faster and faster into the sky.
•
The motor burned out; although the rocket began to slow, it continued to coast ever higher.
•
Eventually, the rocket stopped for a split second before it began to fall back to Earth.
•
Gravity pulled the rocket faster and faster toward Earth until a parachute popped out, slowing its descent.
•
The descent ended as the rocket landed gently on the ground.
4. A story to be told from a graph
Tim, a student at Cumberland Junior High, was determined to ask Caroline for a movie date. Here are the graphs
of his movements from his house to Caroline’s. You write the story.
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Skill Sheet 3.3B
Analyzing Graphs of Motion With Numbers
Speed can be calculated from position-time graphs and distance can be calculated from speed-time graphs.
Both calculations rely on the familiar speed equation: R = D/T.
1. Calculating speed from a position-time graph
This graph shows position and time for a sailboat starting from its
home port as it sailed to a distant island. By studying the line, you can
see that the sailboat traveled 10 miles in 2 hours.
The speed equation allows us to calculate that the vessel speed during
this time was 5 miles per hour.
R = D⁄T
R = 10 miles ⁄ 2 hours
R = 5 miles/hour, read as 5 miles per hour
This result can now be transferred to a speed-time graph. Remember
that this speed was measured during the first two hours.
The line showing vessel speed is horizontal because the speed was
constant during the two-hour period.
2. Calculating distance from a speed-time graph
Here is the speed-time graph of the same sailboat later in the voyage. Between the second and third hours, the
wind freshened and the sailboat increased its speed to 7 miles per hour. The speed remained 7 miles per hour to
the end of the voyage.
How far did the sailboat go during this time? We will first calculate the distance traveled between the third and
sixth hours.
1
Skill Sheet 3.3B Analyzing Graphs of Motion With Numbers
On a speed-time graph, distance is equal to the area between the baseline and the plotted line. You know that the
area of a rectangle is found with the equation: A = L × W. Similarly, multiplying the speed from the y-axis by the
time on the x-axis produces distance. Notice how the labels cancel to produce miles:
speed × time = distance
7 miles/hour × ( 6 hours – 3 hours ) = distance
7 miles/hour × 3 hours = distance = 21 miles
Now that we have seen how distance is calculated, we can consider the distance
covered between hours 2 and 3.
The easiest way to visualize this problem is to think in geometric terms. Find the
area of the rectangle labeled “1st problem,” then find the area of the triangle above,
and add the two areas.
Area of triangle A
Geometry formula
The area of a triangle is one-half the area of a rectangle.
time
speed × ---------- = distance
2
( 3 hours – 2 hours )
( 7 miles/hour – 5 miles/hour ) × ----------------------------------------------- = distance = 1 mile
2
Area of rectangle B
Geometry formula
speed × time = distance
5 miles/hour × ( 3 hours – 2 hours ) = distance = 5 miles
Add the two areas
Area A + Area B = distance
1 miles + 5 mile = distance = 6 miles
We can now take the distances found for both sections of the speed graph to
complete our position-time graph:
2
Skill Sheet 3.3B Analyzing Graphs of Motion With Numbers
3. Practice: Finding speed from position-time graphs
For each position-time graph, calculate and plot speed on the speed-time graph to the right.
1.
The bicycle trip through hilly country.
2.
A walk in the park.
3.
Strolling up and down the supermarket aisles.
3
Skill Sheet 3.3B Analyzing Graphs of Motion With Numbers
4. Practice: Finding distance from speed-time graphs
For each speed-time graph, calculate and plot the distance on the position-time graph to the right. For this
practice, assume that movement is always away from the starting position.
1.
The honey bee among the flowers.
2.
Rover runs the street.
3.
The amoeba.
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Skill Sheet 4.1
Acceleration Problems
This skill sheet will allow you to practice solving acceleration problems. Remember that acceleration is the
rate of change in the speed of an object. In other words, at what rate does an object speed up or slow down? A
positive value for acceleration refers to the rate of speeding up, and negative value for acceleration refers to
the rate of slowing down. The rate of slowing down is also called deceleration. To determine the rate of
acceleration, you use the formula:
speed – Beginning speedAcceleration = Final
-----------------------------------------------------------------------Change in Time
1. Solving acceleration problems
Solve the following problems using the equation for acceleration. Remember the units for acceleration are meters
per second per second or m/sec2. The first problem is done for you.
1.
A biker goes from a speed of 0.0 m/sec to a final speed of 25.0 m/sec in 10 seconds. What is the acceleration
of the bicycle?
25.0
m- – 0.0
m25.0
m---------------------------------------sec
2.5 m
sec
sec
acceleration = ------------------------------------ = ---------------- = ------------2
10 sec
10 sec
sec
2.
A skater increases her velocity from 2.0 m/sec to 10.0 m/sec in 3.0 seconds. What is the acceleration of the
skater?
3.
While traveling along a highway a driver slows from 24 m/sec to 15 m/sec in 12 seconds. What is the
automobile’s acceleration? (Remember that a negative value indicates a slowing down or deceleration.)
4.
A parachute on a racing dragster opens and changes the speed of the car from 85 m/sec to 45 m/sec in a
period of 4.5 seconds. What is the acceleration of the dragster?
5.
The cheetah, which is the fastest land mammal, can accelerate from 0.0 mi/hr to 70.0 mi/hr in 3.0 seconds.
What is the acceleration of the cheetah? Give your answer in units of mph/sec.
1
Skill Sheet 4.1 Acceleration Problems
6.
The Lamborghini Diablo sports car can accelerate from 0.0 km/hr to 99.2 km/hr in 4.0 seconds. What is the
acceleration of this car? Give your answer in units of kilometers per hour/sec.
7.
Which has greater acceleration, the cheetah or the Lamborghini Diablo? (To figure this out, you must
remember that there are 1.6 kilometers in 1 mile.) Be sure to show your calculations.
2. Solving for other variables
Now that you have practiced a few acceleration problems, you can rearrange the acceleration formula so that you
can solve for other variables such as time and final speed.
Final speed = Beginning speed + ( acceleration × time )
speed – Beginning speedTime = Final
-----------------------------------------------------------------------Acceleration
1.
A cart rolling down an incline for 5.0 seconds has an acceleration of 4.0 m/sec2. If the cart has a beginning
speed of 2.0 m/sec, what is its final speed?
2.
A car accelerates at a rate of 3.0 m/sec2. If its original speed is 8.0 m/sec, how many seconds will it take the
car to reach a final speed of 25.0 m/sec?
3.
A car traveling at a speed of 30.0 m/sec encounters an emergency and comes to a complete stop. How much
time will it take for the car to stop if its rate of deceleration is -4.0 m/sec2?
4.
If a car can go from 0.0 to 60.0 mi/hr in 8.0 seconds, what would be its final speed after 5.0 seconds if its
starting speed were 50.0 mi/hr?
2
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Skill Sheet 4.2
Acceleration and Speed-Time Graphs
Acceleration and distance can be calculated from speed-time graphs.
1. Calculating acceleration from a speed-time graph
Acceleration is the rate of change in the speed of an object. The graph below shows that object A accelerated
from rest to 10 miles per hour in two hours. The graph also shows that object B took four hours to accelerate
from rest to the same speed. Therefore, object A accelerated twice as fast as object B.
Clearly, the steepness of the line is related to acceleration. This
angle is the slope of the line and is found by dividing the
change in the y-axis value by the change in the x-axis value.
Acceleration = ∆y
-----∆x
In everyday terms, we can say that the speed of object A
“increased 10 miles per hour in two hours.” Using the slope
formula:
∆y
10 mph – 0 mph
5 mph
Acceleration = ------ = --------------------------------------- = --------------∆x
2 hours – 0 hour
hour
•
Acceleration = ∆y/∆x (the symbol ∆ means “change in”)
•
Acceleration = (10 mph – 0 mph)/(2 hours – 0 hours)
•
Acceleration = 5 mph/hour (read as 5 miles per hour per hour)
Beginning physics students are often thrown by the double per time label attached to all accelerations. It is not so
alien a concept if you break it down into its parts:
The speed changes . . .
. . . during this amount of time:
5 miles per hour
per hour
Accelerations can be negative. If the line slopes downward, ∆y will be a negative number because a larger value
of y will be subtracted from a smaller value of y.
2. Calculating distance from a speed-time graph
The area between the line on a speed-time graph and the baseline is equal to the distance that an object travels.
This follows from the rate formula:
Rate or Speed = Distance
--------------------Time
R = D
---T
Or, rewritten:.
RT = D
miles/hour × 3 hours = 3 miles
1
Skill Sheet 4.2 Acceleration and Speed-Time Graphs
Notice how the labels cancel to produce a new label that fits the result.
Here is a speed-time graph of a boat starting from one place and
sailing to another:
The graph shows that the sailboat accelerated between the second and
third hour. We can find the total distance by finding the area between
the line and the baseline. The easiest way to do that is to break the
area into sections that are easy to solve and then add them together.
A + B + C + D = distance
•
Use the formula for the area of a rectangle, A = L × W, to find
areas A, B, and D.
•
Use the formula for finding the area of a triangle, A = l × w/2, to find area C.
A + B + C + D = distance
10 miles + 5 miles + 1 mile + 21 miles = 37 miles
3. Acceleration from speed-time graph practice
Calculate acceleration from each of these graphs.
1.
2.
3.
2
Skill Sheet 4.2 Acceleration and Speed-Time Graphs
4.
Find acceleration for segment 1 and segment 2.
4. Distance from speed-time graph practice
Calculate total distance from each of these graphs.
1.
2.
3.
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Name:
Skill Sheet 4.3
Acceleration Due to Gravity
One formal description of gravity is “The acceleration due to the force of gravity.” The relationships among
gravity, speed, and time are identical to those among acceleration, speed, and time. This skill sheet will allow
you to practice solving acceleration problems that involve objects that are in free fall.
1. Gravity, velocity, distance, and time
When solving for velocity, distance, or time with an object accelerated by the
force of gravity, we start with an advantage. The acceleration is known to be
9.8 meters/second/second or 9.8 m/sec2. However, three conditions must be
met before we can use this acceleration:
•
The object must be in free fall.
•
The object must have negligible air resistance.
•
The object must be close to the surface of the Earth.
In all of the examples and problems, we will assume that these conditions have
been met and therefore acceleration due to the force of gravity shall be equal to
9.8 m/sec2 and shall be indicated by g Because the y-axis of a graph is vertical,
change in height shall be indicated by y.
Remember that speed refers to “how fast” in any direction, but velocity refers to “how fast” in a specific
direction. The sign of numbers in these calculations is important. Velocities upward shall be positive, and
velocities downward shall be negative.
2. Solving for velocity
Here is the equation for solving for velocity:
final velocity = initial velocity + ( the acceleration due to the force of gravity × time )
OR
v = v 0 + gt
Example:
How fast will a pebble be traveling 3 seconds after being dropped?
v = v 0 + gt
2
v = 0 + ( – 9.8 meters/sec × 3 sec )
v = – 29.4 meters/sec
(Note that gt is negative because the direction is downward.)
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Skill Sheet 4.3 Acceleration Due to Gravity
3. Problems
1.
A penny dropped into a wishing well reaches the bottom in 1.50 seconds. What was the velocity at impact?
2.
A pitcher threw a baseball straight up at 35.8 meters per second. What was the ball’s velocity after
2.5 seconds? (Note that, although the baseball is still climbing, gravity is accelerating it downward.)
3.
In a bizarre but harmless accident, Superman fell from the top of the Eiffel Tower. How fast was Superman
traveling when he hit the ground 7.8 seconds after falling?
4.
A water balloon was dropped from a high window and struck its target 1.1 seconds later. If the balloon left
the person’s hand at –5 meters/sec, what was its velocity on impact?
4. Solving for distance
Imagine that an object falls for one second. We know that at the end of the
second it will be traveling at 9.8 meters/second. However, it began its fall at
zero meters/second. Therefore, its average velocity is half of
9.8 meters/second. We can find distance by multiplying this average velocity
by time. Here is the equation for solving for distance. Look to find these
concepts in the equation:
the acceleration due to the force of gravity × time
distance = ---------------------------------------------------------------------------------------------------------------------- × time
2
OR
1 2
y = --- gt
2
Example: A pebble dropped from a bridge strikes the water in exactly 4 seconds. How high is the bridge?
1 2
y = --- gt
2
1
y = --- × 9.8 meters/sec × 4 sec × 4 sec
2
1
2
y = --- × 9.8 meters/sec × 4 sec × 4 sec
2
y = 78.4 meters
Note that the terms cancel. The answer written with the correct number of significant figures is 78 meters. The
bridge is 78 meters high.
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Skill Sheet 4.3 Acceleration Due to Gravity
5. Problems
1.
A stone tumbles into a mineshaft and strikes bottom after falling for 4.2 seconds. How deep is the
mineshaft?
2.
A boy threw a small bundle toward his girlfriend on a balcony 10.0 meters above him. The bundle stopped
rising in 1.5 seconds. How high did the bundle travel? Was that high enough for her to catch it?
3.
A volleyball serve was in the air for 2.2 seconds before it landed untouched in the far corner of the
opponent’s court. What was the maximum height of the serve?
6. Solving for time
The equations demonstrated in Sections 2 and 3 can be used to find time of flight from speed or distance,
respectively. Remember that an object thrown into the air represents two mirror-image flights, one up and the
other down.
Original equation
Time from velocity
Time from distance
Rearranged equation to solve for time
v–v
t = ------------0g
v = v 0 + gt
1 2
y = --- gt
2
t =
2y
-----g
Try these:
1.
At about 55 meters/sec, a falling parachuter (before the parachute opens) no longer accelerates. Air friction
opposes acceleration. Although the effect of air friction begins gradually, imagine that the parachuter is free
falling until terminal speed (the constant falling speed) is reached. How long would that take?
2.
The climber dropped her compass at the end of her 240-meter climb. How long did it take to strike bottom?
3.
For practice and to check your understanding, use these equations to check your work in Sections 2 and 3.
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