Kinematics Workshop
Academic Success Center
Agenda
● Vector vs Scalar quantities
● SOHCAHTOA
● Working with vectors
● Displacement, speed, etc.
Scalar vs Vector Quantities
● Scalar – magnitude, but no direction.
● Speed, distance, time.
● Vector – magnitude and direction.
● Velocity, displacement, force.
SOHCAHTOA
● Sin = Opposite/Hypotenuse
● Cos = Adjacent/Hypotenuse
● Tan = Opposite/Adjacent
● The hypotenuse lies opposite to the right
angle (90°).
● O stands for the side of the triangle
opposite to .
● A stands for the side of the triangle
adjacent to .
The Components of a Vector
● Any vector can be broken down into its y
and x components.
● Use SOHCAHTOA to find the
components.
● Use Pythagorean theorem to find the
magnitude of a vector.
Addition and Subtraction of
Vectors
● To find the sum of the vectors A and B,
place the tail of B at the tip of A (do not
change direction of either vector).
● Subtracting two vectors is equivalent to
adding the vector being subtracted with
the opposite sign.
● A – B = A + (-B)
● To make vector –B, erase the arrow head
at the tip of B and redraw it where the tail
used to be.
Find the Resultant Vector
● Vector A has a magnitude of 30.0 m and
points in a direction 45º above positive x
axis. Vector B has a magnitude of 50.0 m
and points in a direction of 60º above
negative x axis.
● Find the magnitude and direction of the
resultant vector C.
● Express the vector D = A – C and sketch
the vector.
Find the Resultant Vector
● Resolve vectors A and B into their x and y
components.
● Add together x components to get the x
component of vector C (Cx). Add together y
components to get the y component of vector
C (Cy).
● Use the Pythagorean theorem to find the
magnitude of vector C |C|.
● Find the direction of the resultant vector C by
using tan = Cy/Cx.
● Take the inverse of the tan to find the angle
C makes with the x axis.
Find the Resultant Vector
● Remember to take the negative of vector
C component, when expressing vector D.
Displacement, speed, etc.
● x – displacement. Units – meter.
● V – average speed. Actual distance/Time.
Units – m/s
● V- average velocity. Displacement/Time.
Units – m/s
● a – acceleration. Velocity/time. Units m/s².
Average Speed vs Average
Velocity
● You are taking 290 downtown. The first 5
miles you are moving at an exasperating
speed of 10 mi/hour. You travel the next 7
miles at a 70 mi/hour. What is the average
speed of the entire trip?
Answer
average speed vs average velocity
● 20 mph
A few words about
acceleration…
● It can be negative (deceleration, slowing
down) or positive (acceleration, speeding
up).
● Gravity is also an acceleration. Therefore,
its units are m/s².
● g stands for positive 9.81 m/s². Depending
on the coordinate system you choose to
use, you assign to g (or any other
acceleration) positive or negative value.
● Thus, if you take upward as positive, the
gravity will be expressed with a negative
Units can save you!
● Know the difference
between the
fundamental units –
time (s), length (m),
and mass (kg).
● The derived
quantities are a
combination of
fundamental units.
● Always check units
to make sure you are
on the right track
solving the problem.
● If, for example, your
time is expressed in
meters, it means that
you solved the
problem wrong!
Units can save you!
● Some practice:
● Density = mass/volume. Express volume
and prove the equation with units.
● x = gt²/2. Find the units of x. Show all the
work.
● T = 2p(m½/k½). What are the units of k?
Answer
units
● kg/m3
● ms2/2s2 =m/2
● m/(T/2p)2 =k
Before moving on to the
problems…
● There is not a clear-cut way to solve a
problem in physics. Often, there are
several solutions... Look at it as an
advantage and learn to think through the
problems.
Before moving on to the problems.
Some tips.
● Carefully read the problem.
● State what you are looking for and what you know.
Write it out.
● Sketch the problem. It is not a waste of time! On
the contrary, sketching allows you to think through
the problem and visualize the physical process.
● Come up with a strategy. You may have to try
several strategies before finding a correct one,
and that is OK.
● Find a correct equation. Solve it.
● Watch your units! Is your answer reasonable? Use
units and common sense to check your answer.
Equations of Motion
● V= Vo +at
● X= Xo + Vot+ (at2)/2
● X= Xo + (Vo + V)t/2
● V2= Vo2 + 2a(X-Xo)
One-dimensional kinematics.
Common problems.
● An object has an initial velocity of 1.5 m/s.
It accelerates at a rate of 2.0 m/s² for 5
seconds.
● What is the object’s velocity after that
period of time?
● How far did it travel in this time?
Answer
one dimensional kinematics problem
● 11.5 m/s
● 9.75 m
One-dimensional kinematics.
Common problems.
● A girl throws an apple vertically up in the
air with a velocity of 4.9 m/s. Take g =
9.81 m/s² as your acceleration. What is
the velocity of an apple
● After 0.25s?
● After 1.00s?
● What is the distance and time at which
apple reaches its maximum height?
Answer
apple problem:
● a)2.44 m/s
● b)-4.9 m/s
● c) 1.225 m
One-dimensional kinematics.
Common problems.
● Starting from rest, a jet is catapulted with
a constant acceleration along a straight
line and reaches a velocity of 100 m/s in 3
seconds.
● What is its average acceleration?
● What is the jet’s velocity after it travels a
displacement of 300 m?
One-dimensional kinematics.
Braking time.
● A car is traveling with a constant speed,
when driver applies the brakes, giving the
car a deceleration of 3.0 m/s². The car
stops in a distance of 32 m.
● What was the car’s speed before
deceleration?
● What was its speed in 1sec?
● In 2 sec?
One-dimensional kinematics.
Braking distance.
● You are driving a car at 12 m/s. Suddenly
a deer appears on the road. You slow
down with the acceleration of 3.5 m/s² If
the deer is in the distance of 25 m from
your car
● How close to it do you get?
● How much time does you car need to
come to a complete stop?
One-dimensional kinematics.
Common problems.
● You stand at the top of the building and
throw a ball vertically upward with a
velocity of 12.0 m/s. The ball reaches the
ground 4.75 s later.
● What is the max height reached by the
ball?
● How high is the building?
● With what velocity will the ball reach the
ground?
Two-dimensional kinematics.
Free fall.
● An orange is thrown horizontally from an
open window that is 10.0 m from the
ground level. The orange lands a
horizontal distance of 50.0 m from the
base of the building.
● What was its initial velocity?
One-dimensional kinematics.
Free fall.
● You dive from a high shore into the river
below. If you hit the water 1.3 seconds
later
● How high was the shore?
● What is your velocity when you hit the
water?
Two-dimensional kinematics
● Basically, a combination of two linear
motions – vertical and horizontal.
● Acceleration in x direction is 0 (ax = 0), as we
assume no air resistance.
● As there is no acceleration in x direction, vx
is constant.
● Acceleration in y direction is due to gravity
(ay = g).
● Vy is constantly changing. It has a value of 0
at the highest point. Vy initial has the same
magnitude as vx final, but opposite direction
(parabolic path).
Two-dimensional kinematics.
● X component of the velocity (vx) determines
the range.
● Y component of the velocity (vy) determines
time in air and maximum height.
● It is of no use to memorize equations derived
to determine the range, maximum height,
landing site etc. Besides that it is easy to
make a mistake when memorizing them, your
instructor will want you to show all the work
and understanding of the concepts.
● By using the familiar kinematics equations
and rules of the projectile motion, you will
Two-dimensional kinematics.
Common problems.
● A bullet is fired from a ground level with a
speed of 145 m/s at an angle of 30º above
horizontal. After it has been in the air for 4
sec
● What is its vertical component?
● What is its horizontal component?
Two-dimensional kinematics.
Common problems.
● The same girl as in the aforementioned
problem throws an apple with an initial
velocity of 10 m/s at an angle of 60º with
the ground. We assume no air resistance.
● How long is the apple in the air?
● How far does it go?
Two-dimensional kinematics.
Common problems.
● You throw a stone from the top of the
building upward at an angle of 30º to the
horizontal with the initial velocity of 25.0
m/s. Height of the building is 45.0 m.
● How long does the stone take to reach the
ground?
● Determine the range measured from the
base of the building.
Answer
jet problem
● a)33.3
● b)141.35
Answer
car problem
● a)13.85
● b)10.85
● c)7.85
Answer
deer problem
● a)4.428 m
● b)3.4 s
Answer
building problem
● a)68.6 m
● b) 61.25 m
● c) 38.581 m/s
Answer
orange problem
● 35.7 m/s
Answer
shore problem
● a) 8.2 m
● b) 12.7 m/s
Answer
bullet problem
● a) 125.57 m/s
● b) 33.3 m/s
Answer
apple problem
● a)1.767
● b) 8.83
Answer
stone problem
● a) 4.56 s
● b) 98.724 m