# Free Write for 5 min:

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14 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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Free Write for 5 min:

What is the difference between speed and
velocity? Distance and Displacement?

What is the difference between distance and
displacement?

What does direction have to do with anything?

Chapter 2

Kinematics in One
Dimension

Distance and Displacement

Kinematics

is the branch of mechanics that describes the
motion of objects without necessarily discussing what
causes the motion.

Dynamics

deals with the effect that forces have on motion.

Together, kinematics and dynamics form the branch of
physics known as
Mechanics.

Kinematics

How far have you gone if you
run around the track one time?

Distance vs Displacement

Distance (
d
)

Total length of the path travelled

Measured in meters

scalar

Displacement ( )

Change in position (

x
) regardless of path

x = x
f

x
i

Measured in meters

vector

x

Distance vs Displacement

Distance

Displacement

A

B

Chapter 2

Kinematics in One
Dimension

Speed and Velocity

The total distance traveled divided by the time required to
cover the distance.

time
Elapsed
Distance

speed

Average

SI units for speed:
meters per second

(m/s)

Average Speed

t
d
s

How far does a jogger run in 1.5 hours (5400 s) if his
average speed is 2.22 m/s?

m

12000
s

5400
s
m

22
.
2
d

Example 1:
Distance Run by a Jogger

t
d
s

Δt
s
d

The displacement divided by the elapsed time.

time
Elapsed
nt
Displaceme

velocity
Average

t
t
t
o
o

x
x
x
v
Average Velocity

SI units for velocity:
meters per second

(m/s)

Andy Green in the car
ThrustSSC

set
a world record of 341.1 m/s in 1997.
To establish such a record, the driver
makes two runs through the course,
one in each direction, to nullify wind
effects. From the data, determine the
average velocity for each run.

Example 2

The World’s Fastest Jet
-
Engine Car

s
m
5
.
339
s

4.740
m

1609

t
x
v
s
m
7
.
342
s

4.695
m

1609

t
x
v
Example 2

What is the average velocity for
both runs? (combined)

s
341.1m
4.695)s

(4.740
m

2
*
1609

t
d
s
What is the average speed for
both runs? (combined)

The
instantaneous velocity

indicates how fast the
car moves and the direction of motion at each
instant of time.

t
t

x
v
0
lim

Instantaneous Velocity & Speed

The
instantaneous speed

is the magnitude of the
instantaneous velocity

Chapter 2

Kinematics in One
Dimension

Acceleration

Describe the motion of the ball
as it rolls down the ramp.

What happens to the displacement and velocity
as time goes by?

Draw a Position
vs

time and Velocity
vs

Time
graph of its motion

What if anything changes when the
ramp is facing in the opposite direction?

t
t
t
o
o

v
v
v
a

The difference between the final and initial velocity
divided by the elapsed time

Acceleration

SI units for acceleration:
meters per second per second

(m/s
2
)

What does a negative acceleration
mean?

Scenario

Acceleration

Speeding

up in the positive direction

Slowing down in the positive direction

Speeding up in the negative direction

Slowing down in the negative direction

Question #5

A ball is thrown toward a wall, bounces, and returns to the
thrower with the same speed as it had before it bounced.
Which one of the following statements correctly describes this
situation?

a) The ball was not accelerated during its contact with the wall
because its speed remained constant.

b) The instantaneous velocity of the ball from the time it left the
thrower’s hand was constant.

c) The only time that the ball had an acceleration was when the
ball started from rest and left the hand of the thrower and
again when the ball returned to the hand and was stopped.

d) During this situation, the ball was never accelerated.

e) The ball was accelerated during its contact with the wall
because its direction changed.

Question #6

In an air race, two planes are traveling due east. Plane One
has a larger acceleration than Plane Two has. Both
accelerations are in the same direction. Which one of the
following statements is true concerning this situation?

a) In the same time interval, the change in the velocity of Plane
Two is greater than that of Plane One.

b) In the same time interval, the change in the velocity of Plane
One is greater than that of Plane Two.

c) Within the time interval, the velocity of Plane Two remains
greater than that of Plane One.

d) Within the time interval, the velocity of Plane One remains
greater than that of Plane Two.

e) Too little information is given to compare the velocities of the
planes or how the velocities are changing.

Question #7

Two cars travel along a level highway. An observer notices
that the distance between the cars is
increasing
. Which one of
the following statements concerning this situation is
necessarily

true?

a) Both cars could be accelerating at the same rate.

b) The leading car has the greater acceleration.

c) The trailing car has the smaller acceleration.

d) The velocity of each car is increasing.

e) At least one of the cars has a
non
-
zero

acceleration.

Question #8

The drawing shows the position of a rolling ball at one
second intervals. Which one of the following phrases
best describes the motion of this ball?

a) constant position

b) constant velocity

c) increasing velocity

d) increasing acceleration

e) decreasing velocity

Question #9

At one particular moment, a subway train is moving with a
positive velocity and negative acceleration. Which of the
following phrases best describes the motion of this train?
Assume the front of the train is pointing in the positive
x
direction.

a) The train is moving forward as it slows down.

b) The train is moving in reverse as it slows down.

c) The train is moving faster as it moves forward.

d) The train is moving faster as it moves in reverse.

e) There is no way to determine whether the train is moving
forward or in reverse.

Question #10

Which of the following velocity vs. time graphs
represents an object with a negative constant
acceleration?

Chapter 2

Kinematics in One
Dimension

Constant
Acceleration
Equations

1.

a = acceleration

2.

t = time (elapsed)

3.

v

= final velocity (at time
t
),

4.

v
o

= initial velocity (at time 0)

5.

x

= position (at time t)

6.

x
o

= initial position (at time 0)

AP Kinematic Variables:

t
v
v
a
o

o
v
v
at

at
v
v
o

Equations of Kinematics for
Constant Acceleration

AP Equation

#1

a
v
v
v
v
x
x
o
o
o
2
1
t
v
v
a
o

a
v
v
t
o

a
v
v
x
x
o
o
2
2
2

Equations of Kinematics for
Constant Acceleration

t
v
v
x
x

0
0
2
1

0
2
2
2
x
x
a
v
v
o

AP Equation

#3

at
v
v
o

v
v
t
x
x

0
0
2
1
2
2
1
at
t
v
x
x
o
o

Equations of Kinematics for
Constant Acceleration

v
v
v
o

2
1
t
x
x
v
0

t
v
v
x
x

0
0
2
1
If, a is constant:

t
at
v
v
x
x
o
o
o

2
1
AP Equation

#2

Question #11

Complete the following statement: For an object moving
at constant, positive acceleration, the distance traveled

a) increases for each second that the object moves.

b) is the same regardless of the time that the object moves.

c) is the same for each second that the object moves.

d) cannot be determined, even if the elapsed time is known.

e) decreases for each second that the object moves

A spacecraft is traveling with a velocity of +3250 m/s.
Suddenly the retrorockets are fired, and the spacecraft begins
to slow down with an acceleration whose magnitude is 10.0
m/s
2
. What is the velocity of the spacecraft when the
displacement of the craft is +215 km, relative to the point
where the retrorockets began firing?

Example 8

An Accelerating Spacecraft

ax
v
v
o
2
2
2

d

a

v

v
o

t

+215,000 m

-
10.0 m/s
2

?

+3250 m/s

ax
v
v
o
2
2

m

215000
s
m
0
.
10
2
s
m
3250
2
2

v
s
m
v
2500

s
m
v
2500

Question #12

An object moves horizontally with a constant
acceleration. At time
t

= 0 s, the object is at
x

= 0 m.
For which of the following combinations of initial
velocity and acceleration will the object be at

x

=

1.5 m at time
t =

3 s?

a)
v
0

= +2 m/s,
a

= +2 m/s
2

b)
v
0

=

2 m/s,
a

= +2 m/s
2

c)
v
0

= +2 m/s,
a

=

2 m/s
2

d)
v
0

=

2 m/s,
a

=

2 m/s
2

e)
v
0

= +1 m/s,
a

=

1 m/s
2

Question #13

An airplane starts from rest at the end of a
runway and accelerates at a constant rate. In
the
first second,

the airplane travels 1.11 m.
What is the speed of the airplane at the end of
the
second

second?

a) 1.11 m/s

b) 2.22 m/s

c) 3.33 m/s

d) 4.44 m/s

e) 5.55 m/s

Question #14

An airplane starts from rest at the end of a
runway and accelerates at a constant rate. In
the
first second,

the airplane travels 1.11 m.
How much additional distance will the airplane
travel during the
second

second of its motion?

a) 1.11 m

b) 2.22 m

c) 3.33 m

d) 4.44 m

e) 5.55 m

Chapter 2

Kinematics in One
Dimension

Section 6:

Freely Falling Bodies

In the absence of air resistance, it is found that all bodies

at the same location above the Earth fall vertically with

the same acceleration. If the distance of the fall is small

compared to the radius of the Earth, then the acceleration

remains essentially constant throughout the descent.

This idealized motion is called
free
-
fall

and the acceleration

of a freely falling body is called the
acceleration due to

gravity
.

2
2
2
2
s
ft
30
s
m
10
s
ft
2
.
32
or

s
m
80
.
9
or
g
g

Free Fall

Freefalling bodies

I could give a boring
lecture on this and work
through some examples,
but I’d rather make it
more real…

Use same kinematic equations just substitute
g

for
a

Choose +/
-

carefully to make problem as easy as possible

Free fall problems

The referee tosses the coin up

with an initial speed of 5.00m/s.

In the absence if air resistance,

how high does the coin go above

its point of release?

Example 12

h

a

v

v
o

t

?

-
9.80 m/s
2

0 m/s

+5.00
m/s

h

a

v

v
o

t

?

-
9.80 m/s
2

0 m/s

+5.00 m/s

ah
v
v
o
2
2
2

a
v
v
h
o
2
2
2

m

28
.
1
s
m
80
.
9
2
s
m
00
.
5
s
m
0
2
2
2

h
There are three parts to the motion of the coin. On the way

up, the coin has a vector velocity that is directed upward and

has decreasing magnitude. At the top of its path, the coin

momentarily has zero velocity. On the way down, the coin

has downward
-
pointing velocity with an increasing magnitude.

In the absence of air resistance, does the acceleration of the

coin, like the velocity, change from one part to another?

Conceptual Example 14
Acceleration Versus Velocity

Does the pellet in part
b

strike the ground beneath the cliff

with a smaller, greater, or the same speed as the pellet

in part
a
?

Conceptual Example 15
Taking

Question #19

Two identical ping
-
pong balls are selected for a physics
demonstration. A tiny hole is drilled in one of the balls; and the
ball is filled with water. The hole is sealed so that no water can
escape. The two balls are then dropped from rest at the exact
same time from the roof of a building. Assuming there is no
wind, which one of the following statements is true?

a) The two balls reach the ground at the same time.

b) The heavier ball reaches the ground a long time before the
lighter ball.

c) The heavier ball reaches the ground just before the lighter ball.

d) The heavier ball has a much larger velocity when it strikes the
ground than the light ball.

e) The heavier ball has a slightly larger velocity when it strikes the
ground than the light ball.

Question #20

Two identical ping
-
pong balls are selected for a physics
demonstration. A tiny hole is drilled in one of the balls; and the ball is
filled with water. The hole is sealed so that no water can escape.
Each ball is shot horizontally from a gun with an initial velocity
v
0

from
the top of a building. The following drawing shows several trajectories
numbered 1 through 5. Which of the following statements is true?

a) Both balls would follow trajectory 5.

b) Both balls would follow trajectory 3.

c) The lighter ball would follow 4 and the heavier ball would follow 2.

d) The lighter ball would follow 4 and the heavier ball would follow 3.

e) The lighter ball would follow 4 or 3 and the heavier ball would follow 2
or 1, depending on the magnitude of
v
0
.

Question #21

A cannon directed straight upward launches a ball with an
initial speed
v
. The ball reaches a maximum height
h

in a
time
t
. Then, the same cannon is used to launch a
second ball straight upward at a speed 2
v
. In terms of
h

and
t
, what is the maximum height the second ball
reaches and how long does it take to reach that height?
Ignore any effects of air resistance.

a) 2
h,

t

b) 4
h,

2
t

c) 2
h,

4
t

d) 2
h,

2
t

e)
h, t

Chapter 2

Kinematics in One
Dimension

Section 7:

Graphical Analysis of Velocity
and Acceleration

Calculus

Displacement

Velocity

acceleration

Slope of the line

(derivative)

Area under the curve

(integral)

s
m
4
s

2
m

8

Slope

t
x
Finding velocity

Instantaneous Velocity

2
s
m
6
s

2
s
m

12

Slope

t
v
Finding acceleration

Finding displacement

h
b
w
l
area

2
1
velocity

time

v
o

t

v

v

v
o
= at

at
t
t
v
d
o
2
1

2
0
2
1
at
t
v
d

Question #22

A dog is initially walking due east. He stops, noticing a cat behind him.
He runs due west and stops when the cat disappears into some bushes.
He starts walking due east again. Then, a motorcycle passes him and he
runs due east after it. The dog gets tired and stops running. Which of the
following graphs correctly represent the position versus time of the dog?

Question #23

The graph above represents the speed of a car traveling due
east for a portion of its travel along a horizontal road. Which
of the following statements concerning this graph is true?

a) The car initially increases its speed, but then the

speed decreases at a constant rate until the car stops.

b) The speed of the car is initially constant, but then

it has a variable positive acceleration before it stops.

c) The car initially has a positive acceleration, but then it has a
variable negative acceleration before it stops.

d) The car initially has a positive acceleration, but then it has a
variable positive acceleration before it stops.

e) No information about the acceleration of the car can be
determined from this graph.

Question #24

Consider the position versus time graph shown. Which curve
on the graph best represents a constantly accelerating car?

a) A

b) B

c) C

d) D

e) None of the curves represent a constantly accelerating car.

Question #25

Consider the position versus time graph shown. Which curve on
the graph best represents a car that is initially moving in one
direction and then reverses directions?

a) A

b) B

c) C

d) D

e) None of the curves represent a car moving in one direction then
reversing its direction.