Download The Physical Universe, 10/e Konrad B. Krauskopf, Prof. Emeritus of

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Chapter 2
Motion
2-1
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Chapter 2 Outline: Main Ideas

How scientists describe motion






Constant Acceleration: Gravity
Newton’s Laws


Defining “FAST”: speed
Using VECTORS
Acceleration
Distance, time, acceleration
Defining inertia
Universal Gravitation


An example: circular motion
Newton’s Law of Gravity
2-2
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
What is motion?
We say an object moves if the object changes it’s
position
That is…if the object is at one point in
space at one instant in time, and a
different point in space at another instant
in time.
Everything in the universe is in constant motion.
2-3
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Describing the motion


We know that “motion” means a change in position
The next step is describe the change


SPEED is the term used to describe HOW FAST the location of
any object is changing
Speed is the RATE OF CHANGE OF POSITION
2-4
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
SPEED
The AVERAGE speed is defined:
speed = distance traveled  time spent
v




=
d/t
UNITS of speed: (length / time)
The SI unit of speed is m/s
Speedometer reads Instantaneous Speed
The term RATE is scientific term that generally
refers to a change in any quantity proportional to
time
2-5
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Important Note
 In
a car, the speedometer reads
INSTANTANEOUS SPEED

NOT average speed
 INSTANTANEOUS
SPEED is a
measure of HOW fast at a
particular instant in time
2-6
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Common Units for speed
 Any
distance unit divided by any
time unit is a valid unit for speed
 IN THE SI SYSTEM:
meters per second (m/s)
 Other valid units:
feet per second (f/s)
 miles per hour (mph or mi/h)
 kilometers per hour (km/h

2-7
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Number sense
 10
m/s is about 22 miles per hour
 20 m/s is about 45 miles per hour
 30 m/s is about 67 miles per hour
 40 m/s is about 90 miles per hour
2-8
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Relationships
 The equation for speed can be used to
find out HOW FAR

With knowledge of how fast, and how long
d
v
t
d  vt
d
t
v
v stands for average speed
average speed is distance divided by time
d stands for distance
Distance is average speed multiplied by time
t stands for time
Time is distance divided by average speed
2-9
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
SCALAR VS. VECTOR
SCALAR QUANTITY
A QUANTITY IN WHICH
ONLY MAGNITUDE
IS SPECIFIED
 VECTOR QUANTITY
A QUANTITY IN WHICH
BOTH MAGNITUDE AND
DIRECTION
ARE SPECIFIED

2-10
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Examples

FAMILIAR SCALAR QUANTITIES




TEMPERATURE
(measured in Celsius, Fahrenheit, or Kelvin)
TIME (measured in seconds, minutes, or hours)
MASS
(measured in kilograms or grams)
FAMILIAR VECTOR QUANTITY

VELOCITY
(measured in m/s or mph WITH DIRECTION
SPECIFIED!!)
2-11
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Vector Mathematics
 Since
the vector quantity carries
two pieces of information, it is
often represented by a VECTOR

A straight line with an arrowhead


The length of the line yields information
about MAGNITUDE
The arrow indicates the DIRECTION
2-12
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
2-13
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
2-14
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
VECTOR SUM EXAMPLE

Ordinarily, sum just means simple
addition:
2+2=4

But when it comes to VECTORS, we
must consider the DIRECTION
A
Consider Vectors A and B
B
With a coordinate system reference,
geometry can be used to add the vectors
2-15
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
VECTOR SUM EXAMPLE

We must rearrange them
“HEAD TO TAIL” in order to see where
we end up
B
A
B
B
B
C = A +B B
2-16
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
2-17
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
SO, WHAT IS “ACCELERATION”
Acceleration is
HOW FAST
the velocity CHANGES.
Acceleration is the rate of
change of velocity.
2-18
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Acceleration

Acceleration is
change in velocity
time interval
a


v final  vinitial
t final  tinitial
Velocity is defined by both
SPEED and DIRECTION
This
 means that
Acceleration happens when there is a change
in speed OR a change in direction OR both.
2-19
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Common Acceleration Units
Meters per second per second
2
(m/s )
feet per second per second
2
(f/ s )
miles per hour per hour
2
(mph )
2-20
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
ACCELERATION vs VELOCITY
MOST people are confused about the
difference
 Velocity (or speed) is
how fast your position is changing


Acceleration is
how fast your velocity is changing
2-21
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
How can the velocity change?
 TWO WAYS—
the magnitude can change
(speed up or slow down)
OR
the direction can change
(like when you drive on a curve)
In either case, there is a non-zero
acceleration!
2-22
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Acceleration
 We went from 0 m/s to 27 m/s in 3s
 Acceleration is change in velocity
divided by time
a
v final  vinitial
time
m
m
27  0
m
s
s

9 2
3s
s
This is a “VERY FAST” car!!
2-23
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Notes about acceleration and velocity
Acceleration is a vector
 Acceleration direction can be the SAME
as the direction that the object is moving
or in the OPPOSITE direction



Example? Slowing your car down! In this case,
the direction of acceleration will be considered
“negative”, since it’s effect is to make the
velocity smaller
Velocity is always in the same direction as
the object is moving
2-24
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Finding the final velocity

If we know the initial velocity, acceleration and
time, we can use the above equation to find the
final velocity
a
v final  vinitial
time
and solving for the final velocity :
v final  vinitial  a  time
or
v2  v1  a t
2-25
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Special case: constant acceleration

If we know





Initial velocity
Final velocity
Acceleration
Then we can determine HOW FAR an object
travels in a given amount of time
Recall that distance traveled is equal to the
average velocity multiplied by the time:
d  vt
where v is average velocity
2-26
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Finding HOW FAR

The average velocity is simply
v 2  v1 v 2
v1


v
2
2
2
where v is average velocity
Now we can substitute for final velocity in
terms of accelerati on :
v1  at
v1

v
2
2
2-27
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Finding HOW FAR

Then, we use this average velocity expression in
our equation for d:
d  vt
v 2  v1
a
t
v 2  v1  a t
v2  v1  a t
v1  at
v1

v
2
2
v1 
 v1  at


t
2
2 

2
v1 t at
v1 t



2
2
2
1 2
 v1 t  at
2
2-28
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Example
You are driving your car and approaching
a stop sign.
 You apply the brakes when your
speedometer reads 27 m/s (60 mph)
 If you know that the breaks will provide
an acceleration of a= - 6 m/s2, how far
away must you apply the brakes? The
time it takes to reach the stop sign is 4.5 s
or about 5.0 s.

2-29
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Solution, braking distance
we use our distance equation :
1 2
d  v1 t  at
2
using the values given :
1
2
d  (27) 5  (6)5
2
 135  (75)
 60 m
60 m is approximately 197 feet!
2-30
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Summarize what we know so far:



Speed is the total distance traveled divided by
the total time
Average Velocity is the change in position
divided by total time
Acceleration is the change in velocity divided
by the time taken (two ways)


either change the magnitude of the velocity vector
and/or
change the direction of the velocity vector
2-31
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Now, another special case: FREE FALL
Objects moving under the influence of
only gravity
 Major contributor: GALELEO

2-32
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
2-33
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Galeleos Experiments: dropping the ball
Actually, Galileo
performed experiments
by dropping a stone
 He found that the higher
a stone is when it is
dropped, the larger the
speed of the stone when it
hit the ground.

2-34
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Galeleos Experiments: dropping the ball
Actually, Galileo
performed experiments
by dropping a stone
 He found that the higher
a stone is when it is
dropped, the larger the
speed of the stone when it
hit the ground.

2-35
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Galeleos Experiments: dropping the ball
Actually, Galileo
performed experiments
by dropping a stone
 He found that the higher
a stone is when it is
dropped, the larger the
speed of the stone when it
hit the ground.

2-36
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
GALILEO was a GENIUS!
He realized that the ACCELERATION
acting on the stone is CONSTANT…
 THIS MEANS that the velocity is
DIRECTLY PROPORTIONAL to the
time…

The velocity of a falling object
is proportional to the length of time
that the object has been falling.
2-37
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
What Galileo discovered...
If we neglect air resistance, we see
that
ALL OBJECTS FALL AT THE SAME RATE

The letter ‘g’ is used to represent the constant
value of the acceleration due to gravity (near
the surface of the Earth)
g = 9.8
2
m/s
(English Units: 32 ft/s2)
What does this really mean?
2-38
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Acceleration due to gravity…

The ACCELERATION DUE TO GRAVITY is
approximately 9.8 m/s2
(near the surface of the earth)


This means that gravity causes a change in the
velocity of any object by
9.8 m/s EVERY SECOND
So, drop any object and let it fall toward the
ground:



After 1s , the speed of the object is 9.8m/s
After 2s, the speed of the object is 19.6m/s
Etc.
2-39
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
HEADS UP!!!
Acceleration is a VECTOR
 Acceleration changes VELOCITY
 Acceleration due to gravity

ACTS ONLY
IN THE VERTICAL DIRECTION
2-40
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Well, what does that mean?

Its easy to see if we throw a ball
STRAIGHT UP
 Velocity
is only in UP/DOWN
direction (same as “g”)
 What happens to the velocity?
2-41
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
The ball goes up
We have a situation to consider…The
ball’s direction of travel is in the
OPPOSITE direction of the acceleration,
SO, the speed will DECREASE


Let’s say we gave the ball an initial
upward velocity of about 20 m/s
After 2 s, what will the velocity of the ball
be??
2-42
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Back to “HOW FAR?”

Recall that
1 2
d  v1 t  at
2

But for the dropped object, let d become ‘h’ (to
stand for height), and v1=0 (starts from rest),
with the acceleration a=g (only due to gravity:
1 2
h  gt
2
2-43
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
From this, we can answer “HOW LONG”
 If we know the height from which an
object is dropped, we can use the above
equation to determine HOW LONG
the object will fall:
start with :
1 2
h  gt
2
solve for the time, t :
2h
 t
g
2-44
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Example: motion in one direction
Lots of players brag about
their long
“hang time”.
Some people claim to have
hang times of 2 or 3 seconds...
Now that you know some
physics, how can you show
that this cannot be true?
Hint:
Remember, the distance traveled by an object that is
subject to the downward acceleration of gravity is
described by
1 2
h  gt
2
2-45
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Let’s see how FAR (how high) you could go with a
2 second “hang time”:
2 second hang time means total time from jump to
landing, so let’s say that it took the athlete half the
time to go up, 1 s :
dupward = (1/2) ( 9.8 m/s2) x (t)2
= 4.9 x (1)2
= 4.9 meters
NO WAY!!!
2-46
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

REALITY: Even the good players get at
most a jump height2 feet (0.6 meters)
this gives a “hang time” of
0.6 = (1/2) x g x (t)2
solving this for the time, t gives
t = (0.6/4.9) = 0.35 seconds
Or about 1/3 of a second…..
2-47
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Path of the projectile (TRAJECTORY)

PARABOLIC Trajectory
HIGHEST POINT
Half total time
Half total time
HALF WAY POINT OF
HORIZONTAL DISPLACEMENT
2-48
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
The famous NEWTON’s LAWS


LAW 1
When no net force acts on a body, a body at rest will
remain at rest and a body in motion will remain in
motion. (“LAW OF INERTIA”)
LAW 2
The acceleration of any object is directly proportional to
the NET FORCE on the object and inversely

(a = F / m) proportional to the mass of the object.

LAW3
If one object exerts a force on a second object, then the
second object exerts an equal but oppositely directed
force on the first object (For every action there is a
reaction)
2-49
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
NEWTON2 “The law of motion”

UNIFORM MOTION
constant velocity:
no acceleration (net force =0)
2-50
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
What is a NEWTON??
1 NEWTON = 1 kg x (1m/s2)
mass acceleration
NON-UNIFORM MOTION
velocity is changing due to acceleration
F= m x a
a=F/m
2-51
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Example: Newton’s second law
How much force, in NEWTONS,
is needed to accelerate a truck with a mass
of 3200 kg from a full stop to 60 miles per hour,
if it takes about 10 seconds to reach a speed of 60
miles per hour.
Solution
• To find the FORCE, we need to find the
ACCELERATION
• Acceleration = change in velocity / time
2-52
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Convert everything to metric units
 Truck mass: 3200 kg
 Truck speed: 60 miles per hour- need to
go metric

miles
miles 1609 meters
1 hours
60
 60


hour
hour
1 mile
3600 seconds
m
 27  
s
Force  mass  acceleration
 3200 kg  2.7 m/s2
kg  m
 8640
2
s
2-53
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Action-Reaction

Foot applies force
to a ball
Name the action-reaction
pairs:
One person pulls the hands of a
second person
2-54
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Discussion Example

Horizontal displacement
(ignoring air resistance)
Calculate how far a safety net should be placed
from a human being, being shot from a cannon.
The initial velocity is 40 m/s along the vertical
direction and 2 m/s along the horizontal
direction (Hint, only the vertical velocity will
change)
2-55
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
How long is the daredevil in the air?

Vertical component of velocity is 40 m/s
this is the same as throwing the ball straight up
with a velocity of 40 m/s,
 it will take approximately 4 s to get to the top, and
4 s to get back down: TOTAL TIME = 8 s


IN 8 s, with a horizontal velocity of 2 m/s,
the guy travels
distance = speed x time = 2 (m/s) x 8 (s) = 16 m
2-56
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Projectile Motion NOTES
 The path taken by an object thrown
upward at Earth's surface is a
parabola.
 gravity can only act ONLY on the
VERTICAL component
 HORIZONTAL component
2-57
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
So far, we have been neglectful

We have neglected
• Even though lots of things would not be
possible without it:
•Stopping your car
•Driving your car at all (rolling motion)…
2-58
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
And now, a word from our good pal,
FRICTION
In real life--frictional forces are almost
always present.
 The direction of the friction force
ALWAYS OPPOSES THE MOTION
 Friction is bad: it dissipates energy
 Friction is good: it allows us to roll,
turn, ...and...stop!

2-59