Physical Sciences 2: Lecture 1a
September 1, 2016
Welcome to Physical Sciences 2
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What will we be learning this semester in PS2?
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Physical Sciences 2: Lecture 1a
September 1, 2016
Pre-Reading for Lecture 1a:
Vectors, Velocity, and Momentum
Read this while you wait… much of it may be review; we will cover it briefly in class.
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One of the most important concepts in PS2 is the concept of a vector. You can think of a
vector as being like an arrow—something that has a direction, and a magnitude (or
length). Here are a bunch of vectors, with various magnitudes and directions:
You can move a vector around on the page (or anywhere really) and it will always be the
same vector—as long as it keeps the same magnitude and direction! Here are a bunch of
vectors that are all identical:
Here are some vectors that all have the same magnitude but different directions:
And here are some vectors that all have the same direction but different magnitudes:
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We use vectors to represent physical quantities that require both a magnitude and a
direction. For instance, we use vectors to describe the velocity of an object. In physics,
the word “velocity” means both the speed and direction that an object is moving. So we
need a vector to describe velocity:
The magnitude of a velocity vector tells you the speed of an object:
how fast is it going? (Note that the magnitude is always positive.)
The direction of a velocity vector tells you the direction of motion:
in what direction is it moving?
Speed can be described using any convenient units, like miles per hour or meters per
second. Direction can be specified in many ways—for instance, “north” or “east.”
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Physical Sciences 2: Lecture 1a
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September 1, 2016
One way to specify a vector is by using a coordinate system—typically, a set of x- and
y-axes with a coordinate grid. Here is an example:
A vector points
from the tail
to the head.
It's important to remember that the
vector is the same no matter where it is:
all four arrows drawn on this graph
represent the same vector.
To describe a vector,
you need to tell me how long it is
and in what direction it is pointing.
5 steps
(5, –2)
2 steps
Since it doesn't matter where you place
a vector, if you place its tail at the origin
of your coordinate system, then the
coordinates of the head will be equal
to the components of the vector:
the head here is at coordinates (5, –2)
For this vector, going from the
tail to the head means:
Move 5 steps in the +x direction
and 2 steps in the –y direction
so the components are (x, y) = (5, –2)
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In contrast to a vector, an ordinary number is called a scalar (pronounced SKAY-ler).
We must distinguish vectors from scalars in our equations. If something is a vector, we
will write it with a little arrow above it:

a is a scalar
a is a vector
(In your textbook, vectors are distinguished by boldface type: a is a vector.)

Given a vector v , we will use subscripts to indicate its components:

For the vector v , the x-component is vx, and the y-component is vy.

So in the example pictured above, the vector v has vx = 5 and vy = –2. Note that the
components of a vector are ordinary scalars (numbers)!
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The magnitude of a vector can be found from its components using the Pythagorean
theorem:


The magnitude of v is written as v , and is equal to vx2 + vy2
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Since the magnitude of a vector
is a scalar, we will often use v (without an arrow!) to

represent the magnitude v . So it is extremely important to use the arrow for
vectors—if there’s no arrow, the letter alone means the magnitude of the vector!
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Physical Sciences 2: Lecture 1a
September 1, 2016
Introduction
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Our course philosophy and approach to teaching:
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Learning objectives: After this lecture, you will be able to…
1. Draw arrows (vectors) to represent quantities that have both magnitude and direction.
Give examples of physical quantities that require a vector representation.
2. Describe a vector using its magnitude and angle with respect to an axis, or using its
components in a coordinate system. Convert between these two representations.
3. Use vectors to determine the mean electrical axis of the heart from an EKG.
4. Add or subtract vectors graphically, and using components. Show how unit vectors
can be used to construct a vector from its components.
5. Describe the position of an object using its position vector, and use vector subtraction
to calculate the displacement vector.
6. Calculate the average velocity of an object, and compare average velocity with the
instantaneous velocity.
7. Calculate the average speed of an object, and compare speed versus velocity.
8. Explain what momentum is and why it is important.
9. Find the momentum of a single object or a system of several objects.
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Physical Sciences 2: Lecture 1a
September 1, 2016
Activity 1: Vectors, Angles, and Components
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
The arrow shown below represents the velocity vector ( v ) of a baseball. The length of
the arrow represents how fast the ball is moving (its speed), while the direction of the
arrow represents the direction of the ball.
a)
Draw another arrow representing the same velocity vector, but with the tail of the vector
located at the origin.
b)
What are the x- and y-components of the vector? Does it matter where it is located?
c)
Calculate the magnitude of this vector. (The magnitude is the length of the vector.)
d)
Estimate the standard angle of this vector (defined as the angle measured from the
x-axis, such that the y-axis is defined to have a standard angle of +90°.)
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Physical Sciences 2: Lecture 1a
September 1, 2016
Vectors in Medicine
Vectors and components are crucial tools for analyzing EKG’s…
There are many signals (leads) recorded on
an ECG. Let’s look at two:
Lead I:
(x-component)
Lead aVF:
(y-component)
The large “peak” on the ECG tells about the mean electrical axis of the heart.
When the heart contracts, an
! electrical signal propagates through the heart muscle. The mean
electrical axis is a vector H that points in the direction of this electrical signal.
The two leads
shown above give
the components
of
!
this vector H .
What is the
direction of the
mean electrical
axis for the patient
shown here?
Be careful about
the direction of the
y-axis (aVF)!
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Physical Sciences 2: Lecture 1a
September 1, 2016
Adding and Subtracting Vectors
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  
Suppose I tell you that c = a + b .
What does this mean mathematically?
What does this mean graphically?
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Unit vectors are very useful. Given any coordinate system, we can define:
x̂ : a vector that points in the +x direction with unit length
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ŷ : a vector that points in the +y direction with unit length


Show graphically that a vector with components a = (–2, 3) is equal to a = −2 x̂ + 3ŷ .
  
Now suppose that c = a − b .
What does this mean mathematically?
What does this mean graphically?
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Physical Sciences 2: Lecture 1a
September 1, 2016
Position, Velocity,
and Speed
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1.
2.
3.
This image shows repeated
photographs of a basketball as it
bounces (from left to right).

y
The position vector r is a
vector from the origin of the
coordinate system to the location
of as object. Draw position
vectors for the initial position
and final position of the ball.

The displacement Δr is the
difference betweenthefinal and
initial positions: Δr = rf − ri .
Draw the overall displacement
x
vector between the two positions
from question 1 above.

Δr


The average velocity v during an interval is defined as: v =
Δt
If this photograph was captured at 25 images per second, estimate the average velocity of
the basketball during the duration of this image. (A basketball has a diameter of 23 cm.)
4.
5.
The average speed v of an object is a scalar (not a vector!) defined as the total
distance traveled divided by the total time. Estimate the average speed of the basketball
during the duration of this image.

Bonus! The instantaneous velocity v at any time is defined as the average velocity over
a very short time interval (strictly speaking, in the limit that the interval approaches zero).
Draw some vectors on the image to represent the instantaneous velocity of the ball at
various times.
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y
Physical Sciences 2: Lecture 1a
September 1, 2016
Am I getting it?
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1.
2.
3.
I’m going to start at the origin and walk 6 meters in the +x
direction at a speed of 2 m/s. Then I’ll run back to where I
started at a speed of 6 m/s.

What is my position vector r after walking (before I turn
around)? Choose all that are correct.



a) r = 6 m
b) r = (0 m, 6 m)
c) r = (6 m, 0 m)



d) r = (6 m) x̂
e) r = (6 m) x̂ + (0 m) ŷ
f) r = 6 x̂

g) r = 6 x̂ + 0 ŷ

What is my average velocity v during the entire trip (walking and running back)?
Choose all that are correct.



a) v = 3 m/s
b) v = 0 m/s
c) v = ( 0 m/s ) x̂



d) v = ( 0 m/s ) ŷ
e) v = (3 m/s, 0 m/s)
f) v = (0 m/s, 0 m/s)

g) v = 0
What is my average speed v during the entire trip (walking and running back)?
Choose all that are correct.
a) v = 3 m/s
b) v = 4 m/s
d) v = (3 m/s, 0 m/s)
e) v = 0
c) v = 0 m/s
Bonus! For each of the following equations, write an equivalent equation (or equations!)
involving the vector components. (x and y components will be sufficient)
 

a = 2b + c
K=
1 2
mv
2

1

Fdrag = − Cd ρ A v v
2
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x
Physical Sciences 2: Lecture 1a
September 1, 2016
Activity 2: Motion
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This multi-flash photograph shows two pool balls (6 cm in diameter) rolling on a surface
without friction. The grid lines are spaced 10 cm apart, and the balls are photographed
at a rate of 25 frames per second. Initially, ball 2 is at rest (the dark black one), and ball
1 is entering from the left. The two balls collide and move off to the right.
ball 2
ball 2
ball 1
ball 1
1.

Estimate the velocity ( v ) of:
a) ball 1 before the collision
b) ball 1 after the collision
c) ball 2 after the collision
Activity continues on next page…
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Physical Sciences 2: Lecture 1a
September 1, 2016
Activity 2 (continued)
2.
Identify all the interactions involving these balls.
3.
The mass of a pool ball is about 160 g. Considering the two balls together as a single
system, calculate:

a) the initial momentum pi of the system

b) the final momentum p f of the system
c) Is momentum conserved? Why or why not?
4.
Bonus! The kinetic energy of an object is defined as K = 12 mv 2 . Is kinetic energy
conserved for this system of two balls?
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Physical Sciences 2: Lecture 1a
September 1, 2016
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Physical Sciences 2: Lecture 1a
September 1, 2016
One-Minute Paper
Your name: _____________________________
Names of your group members:
_________________________________
_________________________________
_________________________________
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Please tell us any questions that came up for you today during lecture. Write “nothing”
if no questions(s) came up for you during class.
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What single topic left you most confused after today’s class?
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Any other comments or reflections on today’s class?
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