Motion (Chapter 3)
Aristotle attempted to clarify motion by classification. However, he only
HYPOTHESIS 
 → derived theories and models
PREDICTED 
EXPERIMENT → never did an experiment to test his predictions
Galileo came 2,000 years later and challenged these ideas because he did do the
experiments. Aristotle’s hypothesis & prediction: Heavier objects fall faster than
lighter ones. True or false?
Suppose one drops a Ping-Pong and golf ball simultaneous. According to Aristotle, since
the golf ball is 10 times heavier then the Ping-Pong, the golf ball falls a distance 10
longer than the Ping-Pong does in the same amount of time. That is, if a golf ball has
fallen 10 m, then a Ping-Pong would have fallen 1 m in the same amount of time.
Galileo came 2,000 years later and challenged these ideas by way of experiments.
DEMO Drop a Ping-Pong and golf ball simultaneously
Galileo did similar experiments involving incline planes and found it to be FALSE. As a
result, Aristotle’s theories on motion were invalid.
DEMO Roll different size balls down an incline
Representing Motion: Dot Physics & Vectors
I do not have to tell you about motion – you already have the intuition about distance,
velocity, and acceleration. My goal is to put motion into the language of physics.
SPEED ≡ the measure of how fast something is moving
where ≡ is a definition or defined symbol. I don’t need to tell you this. When you've been
driving faster than what is posted on the highway, and suddenly notice a patrolman
behind you, the little hairs on the back of your neck start standing-up because you know
that your speed was too high! Equationwise, I will write as
SPEED =
distance
time
= distance/time
When you look at your speedometer it reads
mph =
miles
hour
≡
distance
time
In this course we use the metric system because it is more convenient for doing
calculations. In the language of dimensional analysis, we write
distance
meter
m
=
[speed] = =
time
second s
Physics is the science of pictures, and so we draw pictures to describe motion using
quantities called vectors. Vectors are arrows that (i) length and (ii) point in a particular
direction. Although we just talked about speed, the correct expression for speed is
velocity; velocity is a more complete description. Velocity is a vector (has both length
and direction) while speed is only represented by the length of the velocity vector.
Different Kinds of Speed Definitions
Currently, I’m living in Watsonville. When I drive my car to Cabrillo, it can be stop and go
depending on the traffic. It does not make much sense to talk about my speed at anyone
point because sometimes (unbeknown to me) traffic eases up and at other times, gets
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really bad. My speed could be 65 mph or 10 mph on Highway 1. It is better to think in
terms of average speed. vave accounts for slow and fast driving over a time interval and
averages it out.
The kind of speed we are most interested in this course is instantaneous speed:
Instantaneous speed ≡ v =
the speed at any one instant
Examples
• The speed you read on your speedometer while driving at any one instant
• Radar guns are used to measure the vinst of baseballs thrown by baseball pitchers
CHECK QUESTION
Is a police officer interested in vavg or vinst?
VELOCITY
When we talk about speed we ask “how fast.” When one talks about velocity, we ask two
questions: “how fast” & “which way.”
v ≡ VELOCITY has SPEED and DIRECTION
• The length of the vector is called the which is the speed
• The DIRECTION tells us which way the object is pointing
There are several types of velocities that we will consider.
1. Constant velocity: constant speed and constant direction
Example: driving at a speed of 65 mph and traveling in a straight line
(v = 65 mph, East)
Objects traveling at constant velocity will travel equal distances between two
consecutive time points (say 1 second intervals). The easiest way to see this is by
drawing a vector diagram of constant velocity. Since we have constant speed, the
length of the vectors must all be the same; constant
direction implies the vectors must all point in the same
direction, as indicated in the picture.
constant speed, constant direction
2. Changing velocity: either the speed changes or the direction changes (so there are
two possibilities) (i) const direction, changing speed and (ii) const speed, changing
direction
(i) Change in speed & constant direction (speeding up or
slowing down while driving straight). This happens on the
highway: the driver next to you senses that you’re about to
change lanes.
changing speed, constant direction
(ii) Change in direction & constant speed (turning while driving at a constant
speed). When one drives North on highway 1 towards Santa Cruz, you
encounter the “fish hook.” If you are an upstanding citizen you will be driving the
fixed speed 20 mph BUT constantly changing your direction in order to stay on
the road.
changing direction, constant speed
(Aside: Change speed and direction; encounter with “Big Moody
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Examples
1. A jogger running east at a steady pace suddenly develops a cramp. He is lucky: A
westbound bus is sitting at a bus stop just ahead. He gets on the bus and enjoys a
quick ride home. Draw a motion diagram of the jogger for the entire motion described
here.
Solution: Despite the detail of the problem we must still make some assumptions.
For example, does the jogger slow much to get on the bus? Is the bus an express
line with no stops until the jogger’s home? The simplest case would be to assume
the jogger does not slow much before getting on the bus, and yes, the bus is an
express line with no stops until the jogger’s home.
2. Explain the motion diagram using short concise sentences.
Solution: A person walks steadily at a constant speed along a path that turns from
north towards the west and continues directly west.
CHECK QUESTION
Why a car’s gauge is called a speedometer and not a velociometer?
ACCELERATION: Change in Velocity
Every time one changes their velocity, acceleration has occurred. Since velocity is a
vector, acceleration must also be a vector (has both magnitude and direction). One has
an intuitive feel for how small or large the change in velocity is. There are 3 ways to
change your velocity (or accelerate): braking – move forward; speeding up – move
backwards; and turning – pushed to the side.
Magnitude: when acceleration occurs your body feels some kind of pushing or pulling
action (“force”). If you are braking, pushing on the brake-petal softly causes you're
velocity to decrease slowly and the result is your body moves forward; however, if you
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slam the brakes, you change your velocity quicker and feel this forward motion
stronger. That is, a greater change in velocity implies a higher acceleration – this is
what is meant by the magnitude of the acceleration. Same idea applies to speeding up
or turning motion.
Direction: the direction of the acceleration is determined by the direction of your body.
When one speeds up, your body is "pushed" in the opposite direction of the velocity. In
this case, the direction of the acceleration is opposite of the direction of your body.
Therefore, acceleration and velocity when speeding up are in the same direction.
Applying this same reasoning to slowing down, when braking your body moves forward,
so the acceleration is opposite of my body's motion (inertia) so the acceleration is
opposite to the velocity.
Challenge: we are turning – which direction does your body move? What is the direction
of the acceleration? When making a turn, your body moves outwards from the
turn, so the direction is inwards.
Acceleration is the rate at which one changes velocity and is write as
 

v − v 0 ∆v
Change in Velocity

=
→ a =
Acceleration =
t − t0
∆t
time interval
Similar to average velocity, we plot the velocity as a linear equation:
∆v
=
a

→ v(t)
= v 0 + at
∆t
Conceptually, acceleration is more difficult to comprehend and fairly common to mix up
velocity with acceleration. Imagine you are on a highway ramp and wish to enter traffic
when there many cars. When you push the petal to accelerate (my old car was such a
lemon – pushing the gas petal or accelerator did not necessarily create acceleration)
your car begins to change speed as follows:
My change in velocity is
Units: [a] = [ ∆v/∆t ] =
m/s2
Example
Suppose you are riding your bike at 20 m/s and you approach a steep hill. Draw the
motion diagram as the bike decelerates at 5 m/s2.
=
a
∆v 5 m/s
=
= 5 m/s2 ≡ constant
∆t
1s
Extending Motion Diagrams to Include Acceleration
Quick review of velocities and vectors
1. Constant velocity:
change speed but constant direction
2. Changing velocity: 
constant speed but change direction
Picture wise, I see the following:
• Speeding up or slowing
• Turning at constant speed:
down:
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What does it mean to accelerate in terms of vectors? We need Newton’s
laws to completely explain how acceleration changes velocity; however, we
can easily see why it does it without going into those details. The spring
attached to this cart will indicate the direction of acceleration (since force ∝
acceleration).
DEMO
Pasco Carts with attached spring
CIRCULAR MOTION
I want to expand on the ideas of circular motion and its acceleration.
A particle is in uniform circular motion if it travels around a circle or a circular arc at
constant (uniform) speed. Although the speed does not vary, the particle is accelerating.
That fact may be surprising because we often think of acceleration (a change in velocity)
as an increase or decrease in speed. However, since velocity is a vector, when the
speed = constant and the direction is changing, one has uniform circular motion.
Remember that you will feel some sensation when you accelerate – going around a
curve you feel something pushing you towards the door.
DEMO
Flat plate demo
• Just like the spring acting on the cart, the string forces the plate to move in a circle
by applying acceleration that is always perpendicular to the velocity. The
acceleration is always directed radially inward and this type of acceleration
associated with uniform circular motion is called centripetal (center seeking)
acceleration.
DEMO
String with paper bob for circular motion
• The velocity is always directed tangent to the circle in the direction of motion.
• If the time to make one revolution is fixed, note that the bigger the circle of travel,
the greater the velocity of the bob.
FREE FALL
As stated earlier, Aristotle attempted to clarify motion by classification. However,
he only
Hypothesis 

Predicated  ⇒ never performed a single expt to test his ideas

Experiment 
Galileo came 2,000 years later and challenged these ideas because he did do
experiments. Aristotle claimed that heavier objects fell faster than lighter ones.
DEMO Drop a heavy and light object simultaneously.
Many people have heard about the famous experiment where Galileo dropped
two objects (lead and wood) of the leaning Tower of Pisa - this clearly untrue.
What is certain is that Galileo used “frictionless” incline planes to prove his ideas.
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When objects are dropped, they clearly hit the ground due to the force of gravity.
The force of gravity effects all objects that have mass. (If time, comment on the
difference between Newtonian and GR.) When air resistance is neglected, we
call it FREE FALL. A strange thing at gravity is that it is independent of mass
and therefore, all objects will hit the ground at the same time when either
• Two objects are dropped from rest (e.g., my son and a Sumo wrestler)
• Two objects are thrown down at the same speed.
ALL OBJECTS accelerate towards the earth at a constant rate of acceleration
a= g= 9.81 m/s2
Experimental Evidence
Apollo 15 astronaut David R. Scott dropped a hammer and a feather from waist
high and struck the lunar surface simultaneously. Also, Steve Chue of Stanford
University (now the director LBL) measured the free fall of two different kinds of
atoms and showed that they fall at g.
Vector Review of Horizontal motion: speeding up and slowing down
If objects are close to the surface of the earth, the acceleration is constant and
points towards the center of the earth. So objects that are moving against the
acceleration due to gravity, clearly slow down, whereas moving with the
acceleration, clearly speed-up. The sign conventions for the acceleration
(negative pointing down) and velocity (positive for up and negative for down) are
shown in the diagram.
What does this mean? For the moment, let’s round off to 10 m/s so that it acts
like $10-bills.
Check Question
Suppose a rifle was fired straight downwards from a high-altitude balloon with
a muzzle speed of 100 m/s. What is the ACCELERATION of the bullet after 1
sec? What is the speed after 3 s?
Point of Interest - Dropping a baseball from very high heights.
In 1939, Joe Sprinz of the San Francisco Baseball Club attempted to break the record
for catching a baseball dropped from the greatest height. Members of the Cleveland
Indians had set the record the preceding year when they caught baseballs dropped
about 700 ft from atop a building. Sprinz used a blimp at 800 ft.
How long was the ball in the air? 7.1 sec
What is the velocity of the ball just before it is caught? -226 ft/s = -154 mph
If air resistance was included, it would reduce the velocity. However, the speed must
have been considerable, because when Sprinz finally managed to get a ball in his glove
(on the 5th try), the impact slammed the glove and hand into his face, fracturing the
upper jaw in 12 places, breaking five teeth, and knocking him unconscious. And he
dropped the ball.
70 m / s =
157mph (with no air resistance)
( 7 s ) ⋅ (10 m / s2 ) =
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