Unit 1 - Kinematics

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Unit 1
-

Kinematics

1

Unit 1
-

Kinematics

Kinematics

Kinematics is the branch of mechanics that
describes the motion of objects. In kinematics, we describe the

Newton’s Laws, we’ll discuss why an object moves.

Day 1: 1
-
dimensional constant speed motion

Particle

Our objects have the follow
ing properties:

Position (x)

Definition
:

SI unit:

Distance (d)

Definition:

SI unit:

Displacement (Δx)
=
䑥ai湩ti潮:
=
=
=
b煵慴a潮:
=
=
p䤠f湩琺
=
=
Delta (

)

Meaning:

Practice Problem

A particle moves from x = 1.0 m to x =
-
1.0 m.

What is the distance

d traveled by the particle?

What is the displacement of the particle?

Practice Problem

You get on a
F
1
0 meters at the bottom. When you reach the top on the first rotation

what distance have you traveled?

what is your

displacement from the bottom?

Practice Problem

You are driving a car on a circular track of diameter 40 meters. After you have driven around 2 ½ times, how far
have you driven, and what is your displacement?

2

Unit 1
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Kinematics

Practice Problem

If

x is the displacement of a particle, and d is the distance the particle traveled during that displacement, which of
the following is always a true statement?

a) d = |Δx| b) d < |Δx|

c) d > |Δx|

d) d
>

|Δx| e) d
<

|Δx|

Explain:

Aver
age Speed

Definition:

Equation:

SI unit:

Average Velocity

Definition:

Equation:

SI unit:

Practice Problem

A car makes a trip of 1½ laps around a circular track of diameter 100 meters in ½ minute. For this trip

what is the average
speed of the car?

what is its average velocity?

Practice Problem

How long will it take the sound of the starting gun to reach the ears of the sprinters if the starter is stationed at the
finish line for a 100 m race? Assume that sound has
a speed of about 340 m/s.

Unit 1
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Kinematics

3

Graphical problem

Describe the motion of this particle.

Describe the motion of this particle.

What physical feature of the
position v time
graph gives
indicates

constant velocity?

_
___________________________________________________________________
______________________
___

Graphical problem

Determine the average velocity from the graph.

Does this graph represent motion at constant velocity?
Why or why not?

Can you determine average velocity from the time at
point A to the time at point B from this graph?

Graphical problem

Determine the average velocity between 1 and 4 seconds.

Practice problem

You drive in a
straight line at 10 m/s for 1.0 hour, and then at 20 m/s for 1.0 hour. What is your average velocity?

Practice problem

You drive in a straight line at 10 m/s for 1.0 km, and then you drive in a straight line at 20 m/s for another 1.0 km.
What is your
average velocity?

t

x

A

B

x (m)

t

x

t

x

4

Unit 1
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Kinematics

Instantaneous Velocity

Definition:

Graphical determination of instantaneous velocity

Draw a tangent line to the curve at B. The slope of this line gives the
instantaneous velocity at that
specific time.

Graphical problem

Determine the instantaneous velocity at 1.0 second.

t

x

B

Unit 1
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Kinematics

5

In the position vs. time graphs below, all the times are in seconds (s), and all the positions are in meters (m).
Rank these graphs on the basis of whic
h graph indicates the greatest displacement from beginning to end of
motion. Give

the highest rank to the one(s)

with

the greatest displacement, and

give the lowest rank to the one(s)
indicating the least displacement. If two graphs indicate the same displ
acement, give them the same rank. Note:
Zero is greater than negative, and ties are possible.

Greatest

1)_______

2)______

3)______

4)_______

5)______

6)______
Least

Rank these spheres on the

basis of the greatest displacement over the
first 3 seconds. Give the highest rank to
the one(s)

with the

greatest displacement, and give the lowest

rank to the one(s)

indicating

the
lowest
displacement.

If

two

motion diagrams indicate the same displacement for the 3
-
second interval, give them the
s
ame rank.

Highest 1________ 2________ 3________ 4________ 5________ 6________ Lowest

Accelerated Motion

6

Unit 1
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Kinematics

Acceleration is:

Equation for constant (uniform) acceleration:

SI Unit:

What does the sign of the acceleration mean?

Positive acceleration:

Negative acceleration:

A word about language: It is useful to avoid the words
increases

and
decreases

when describing changes to
quantities that can be negative, like acceleration and velocity and position. Instead, use the phrases:

becomes more positi
ve/less negative

becomes less positive/

more negative

If you encounter something like
the velocity decreases
, it means becomes more negative.

When referring quantities that have no negative meaning,
like speed and distance, it is perfectly okay to say
increases and decreases.

Standard units:

Another way to think about the units:

Other useful acceleration units:

What acceleration units would an engineer who designs mot
orcycles for market in the United States use?

Europe?

Acceleration has a sign. What does it mean?

The following data show the velocity of for two different objects, A and B, taken at 1 second intervals.

A

0s

1s

2s

3s

4s

5s

6s

-
3m/s

-
2m/s

-
1m/s

0m/s

1m/s

2m/s

3m/s

The acceleration of A is:

B

0s

1s

2s

3s

4s

5s

6s

6m/s

4m/s

2m/s

0m/s

-
2m/s

-
4m/s

-
6m/s

The acceleration of B is:

What rule about the sign of acceleration and velocity will tell me whether an object moving in a straight line is
speeding up or slowing down?

An object speeds up when:

An object slows down when:

Practice Problem

A 747 airliner reaches its takeoff speed of 180 mph in 30 seconds. What is its average acceleration?

The eight situations below show before and

after "snapshots" of a car's velocity. Rank these situations, in

terms of
the change in velocity, from most positive to most negative. All cars

have

the same

mass

and

they traveled the
same distance. Negative numbers, if any, rank lower than positive ones

(
-
20 m/s <

-
10

m/s < 0< 5).

Unit 1
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Kinematics

7

Most Positive

1________ 2________ 3________ 4________ 5________ 6________
Least

Graphical Problem

Describe the motion of
objects depicted in these velocity v time graphs.

What physical feature of the graph gives the acceleration?

Practice Problem

Determine the acceleration from the graph.

Position v Time Graphs of Accelerating Objects

Cart on a Ramp

The
ball

is
released fro
m the top
. The positive direction is to the right.

t

v

t

v

t

v

8

Unit 1
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Kinematics

Position v Time Graphs of Accelerating Objects

Cart on a Ramp

The
cart

is launched up the ramp from the bottom.

It rolls to the top, then stops.

The positive direction is to the
right.

Draw Graphs for Stationary Particles

Draw Graphs for Constant Velocity

time

postion

time

velocity

time

acceleration

time

postion

time

velocity

time

acceleration

time

postion

time

velocity

time

acceleration

time

postion

time

velocity

time

acceleration

Unit 1
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Kinematics

9

Draw
g
raphs for
c
onstant
n
on
-
zero
a
cceleration

time

postion

time

velocity

time

acceleration

time

postion

time

velocity

time

acceleration

time

postion

time

velocity

time

acceleration

10

Unit 1
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Kinematics

Rank these
spheres on the

basis of the greatest displacement over the first 3 seconds. Give the highest rank to
the one(s)

with the

greatest displacement, and give the lowest

rank to the one(s)

indicating

the
lowest
displacement.

If

two

motion diagrams indicate the s
ame displacement for the 3
-
second interval, give them the
same rank.

Rank each case from the highest to the lowest displacement based on the first three seconds. Note: Zero is
greater than negative, and ties are possible.

(8b)

Highest

1________ 2_______
_ 3________ 4________ 5________ 6________
Lowest

In the position vs. time graphs below, all the times are in seconds (s), and all the positions are in meters (m).
Rank

these

graphs

on

the basis

of which

graph indicates

the

greatest

average

speed,

where

the average speed is
calculated from the beginning to the end of motion. Give the highest rank to the one(s) with the greatest average
speed, and give the lowest rank to the one(s) indicating the least average speed. If two graphs indicate the same
average

speed, give them the same rank.

(8c)

Greatest

1________ 2________ 3________ 4________ 5________ 6________
Least

Unit 1
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Kinematics

11

The following drawings indicate the motion of a ball subject to one or more forces on various surfaces. Each circle
represents the
position of the ball at succeeding instants of time; the ball marked “t=0” indicates its initial position.
Each time
-
interval between successive positions is equal. The coordinate system is parallel to the ball’s motion,
with motion to the right as positiv
e.

Rank each case from the highest to the lowest velocity based on the ball’s last velocity. Note: Zero is greater than
negative, and ties are possible.

(8d)

Highest

1________ 2________ 3________ 4________ 5________ 6________
Lowest

Rank each case fro
m the highest to the lowest velocity based on the ball’s last velocity. Note: Zero is greater than
negative, and ties are possible.

(8e)

Highest

1________ 2________ 3________ 4________ 5________ 6________
Lowest

12

Unit 1
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Kinematics

Kinematic Equations

Equation 1:

U
se this one when you aren’t

Equation 2:

Use this one when you aren’t

Equation 3:

Use this one when you aren’t

Practice Problem

On a ride called the Detonator at
Worlds of Fun

in Kansas City, passengers accelerate straight downward from 0
to 45 mph in 1.0 second.

What is the average acceleration of the passengers on this ride?

How fast would they be going if they accelerated for an additional second at this rate?

Sketch approximate x
-
vs
-
t, v
-
vs
-
t and a
-
vs
-
t graphs for this ride.

Practice Problem

A horse is running with an initial velocity of 11 m/s, and begins to accelerate at

1.81 m/s
2
. How long does it take
the horse to stop?

Practice problem

A plane is flying in a northwest direction when it lands, touching the end of the runway with a speed of 130 m/s. If
the runway is 1.0 km long, what must the acceleration of the plane be if it is to stop while leaving ¼ of the runway
remaining as a safety
margin?

Unit 1
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Kinematics

13

Practice Problem

You are driving through town at 12.0 m/s when suddenly a ball rolls out in front of you. You apply the brakes and
decelerate at 3.5 m/s
2
.

How far do you travel before stopping?

When you have traveled only half the
stopping distance, what is your speed?

How long does it take you to stop?

Sketch approximate x
-
vs
-
t, v
-
vs
-
t, a
-
vs
-
t graphs for this situation.

14

Unit 1
-

Kinematics

Consider the position vs. time graph below for cyclists A and B.

Do the cyclists
start at the same point? How do you know? If not,

At t= 7s, which cyclist is ahead? How do you know?

Which cyclist is travelling faster at t = 3s? How do you know?

Are their velocities equal at any time? How do you know?

What is happening at the intersection of lines A and B?

Consider the new position vs. time graph below for cyclists A and B.

How does the motion of the cyclist A in the new graph compare to that of A in
the previous graph?

How does the
motion of cyclist B in the new graph compare to that of B in the
previous graph?

Which cyclist has the greater speed? How do you know?

Describe what is happening at the intersection of lines A and B.

Which cyclist traveled a greater distance du
ring the first 5 seconds? How do you know?

Unit 1
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Kinematics

15

Free Fall

Describe:

What is the acceleration in free fall?

Which direction does the acceleration point?

Symmetry in Free Fall

How is time allocated when an object is thrown straight up and returns
to the thrower?

How does initial velocity compare with final velocity when an object is thrown straight up and returns to the
thrower?

Practice Problem

You drop a ball from rest off a 120 m high cliff. Assuming air resistance is negligible,

how lon
g is the ball in the
air?

W
hat is the ball’s speed and velocity when it strikes the ground at the base of the cliff?

W
hat is the ball’s speed and velocity when it has fallen half the distance?

S
ketch approximate x
-
vs
-
t, v
-
vs
-
t, a
-
vs
-
t
graphs for this situation.

16

Unit 1
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Kinematics

Practice Problem

You throw a ball straight upward into the air with a velocity of 20.0 m/s, and you catch
it at the same height of
release
.

How long is the ball in the air?

How high does the ball go?

What is the
ball’s velocity when you catch it?

Sketch approximate x
-
vs
-
t, v
-
vs
-
t, a
-
vs
-
t graphs for this situation.

Pretest Free Response

Case 1:

Ball A is dropped from rest at the top of a cliff of height
h

as shown. Using
g

as the acceleration due to
gravity, derive an expression for the time it will take for the ball to hit the ground.

Case 2:

Ball B is projected vertically upward from the foot of the cliff with an initial speed of
v
o
. Derive an
expression for the maximum height
y
max

reached by the ball.

B

h

v
o

A

h

Unit 1
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Kinematics

17

Case 3
: Ball A is dropped from rest at the top of the cliff at exactly the same time Ball B is thrown vertically
upward with speed
v
o

from the foot of the cliff such that Ball B will collide with Ball A. Derive an expression for the
amount of time that will elapse before they collide.

Case 4
: Ball A is dropped from rest at the top of the cliff at exactly the same time Ball B is

projected vertically
upward with speed
v
o

from the foot of the cliff directly beneath ball A. Derive an expression for how high above the
ground they will collide.

Pinewood Derby Problem

The position of a pinewood derby car was observed at various

times. The results are summarized in the table
below.

x(m)

0

2.3

9.2

20.7

36.8

57.5

t(s)

0

1.0

2.0

3.0

4.0

5.0

Calculate the average velocity of the car for the first second.

Calculate the average velocity of the car for the last three seconds.

Calculate the average velocity of the car for the entire five seconds.

On your own graph paper, do the following. Attach this work to the end of this packet.

On graph paper, draw a position
vs

time graph for the car using the dat
a

in the table

above.

vs

time graph, draw tangent lines at three different points on the curve. Measure the slope of
these tangent lines and determine the instantaneous velocity at all three points.

On a separate sheet of graph paper, draw a velocity
vs

time graph using the instantaneous velocities you
obtained in the step above, and the times at which those instantaneous velocities occur.

vs

time graph, determine the acceleration of the car.

18

Unit 1
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Kinematics

A Brief Introduction to Vectors

Vector

Examples

Scalar

Examples

This is how you draw a vector…

Direction of Vectors

Vector direction is the

direction of the arrow, given by
an angle
.

Vector angle ranges

W
hat we’ll do with vectors:

Throughout the year, we’ll
need to convert between vector form (magnitude and direction) and coordinate form (
x

and
y

values). These processes are called decomposition and reconstitution.

Decomposition

When given an ang
le

Θ

r
, the
x

and
y

components are given by the following:

Wh敮⁧ v敮⁸⁡湤 y val略s,⁦i湤⁴ 攠e慧湩瑵摥

r
,

and
direction
,

Θ
,

by the following
:

(

)

Ex慭灬攠ef⁤ c潭灯si瑩潮:

A⁢ ll⁩s⁴ row渠慴a愠ap敥d 潦‱ ⁭⽳⁡ ⁡ ⁡ gl攠潦″ °⸠䙩湤⁴ e⁸⁡ d y⁣潭灯湥湴n⁯ ⁴ 攠vel潣ity⸠

V散瑯爠t潲o:

㔠5琠㔳
°

䍯潲Ci湡t攠e潲o:

(㌬3㐩

r散潮s瑩瑵tion

Unit 1
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Kinematics

19

Example of reconstitution:

You are following a treasure map which takes you from your camp, 2 km east and then 3 km north. What is the
magnitude and direction of your displacement when you’re arrive at the
spot marked x
?

Sample problem

A surveyor stands on a riverbank directly

across the river from a tree on the opposite bank. She then walks 100 m
downstream, and determines that the angle from her new position to the tree on the opposite bank is 50
o
. How
wide is the river, and how far is she from the tree in her new location?

Sample problem

You are standing erect at the very top of a tower and notice that in order to see a manhole cover on the ground 50
meters from the base of the tower, you must look down at an angle 75
o

below the horizontal. If you are 1.80 m tall,
how high is the tower?

You are driving up a long inclined road. After 1.5 miles you notice that signs along the roadside indicate that your
elevation has increased by 520 feet.

a) What is the angle of

b) How far do you have to drive to gain an additional 150 feet of elevation?

Sample Problem

Find the x
-

and y
-
components of the following vectors

R

= 175 meters @ 95
o

v
= 25 m/s @
-
78
o

a

= 2.23 m/s
2

@ 150
o

20

Unit 1
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Kinematics

A

and
B

graphically by drawing them together in a head
to tail arrangement.

Draw vector
A

first, and then draw vector
B
such that its tail is on the
A
.

Then draw the sum, or resultant vector, by dra
wing a vector from the
tail of
A

B
.

Measure the magnitude and direction of the resultant vector.

Definition of Resultant:

Unit 1
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Kinematics

21

Two Dimensional Kinematics

Horizontal Shot

When studying two dimensional kinematics, we
come to understand motion as a combination of a horizontal
constant velocity and a vertical acceleration.

Therefore, when analyzing motion, we use our three kinematics equations, separated into x and y components.

Sample Problem

Horizontal Launch

A ball rolls off a table 75 cm high table at a speed of 3 m/s. How long does it take the ball to hit the
ground?

The same ball rolls off the same table, now with a speed of 6 m/s. How long does it take the ball to hit the
ground?

Referring back to the first ball, what is it’s velocity (magnitude and direction)
0.2 seconds after leaving the table?

horizontal constant
velocity

vertical
accelerated
motion

22

Unit 1
-

Kinematics

Sketch position, velocity, and acceleration v time graphs for the horizontal and vertical directions.

Horizontal Motion

Vertical Motion

Sample Problem

The Zambezi River flows over Victoria Falls in Africa. The falls are approximately 108 m high. If the river is flowing
horizontally at 3.6 m/s just before going over the falls, what is the speed of the wa
ter when it hits the bottom?
Assume the water is in freefall as it drops.

a

a

v

v

x

y

Unit 1
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Kinematics

23

Sample Problem

Playing shortstop, you throw a ball horizontally to the second baseman with a speed of 22 m/s. The ball is caught
by the second baseman 0.45 s later. How far were
you from the second baseman?

What is the distance of the vertical drop?

24

Unit 1
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Kinematics

2 Dimensional Kinematics

Non Horizontal Shot

In the case where the object is projected at an upward or downward angle, we must first decompose the initial
velocity
into its horizontal and vertical components.

Then proceed to work problems just like you did with the zero
launch angle problems.

Snowballs are thrown with a speed of 13 m/s from a roof 7.0 m above the ground. Snowball A is thrown straight
downward; snow
ball B is thrown in a direction 25
o

above the horizontal. When the snowballs land, is the speed of
A greater than, less than, or the same speed of B? Verify your answer by calculation of the landing speed of both
snowballs.

Projectiles
launched over level ground

These projectiles have highly symmetric characteristics of motion.

Sample problem

A soccer ball is kicked with a speed of 9.50 m/s at an angle of 25
o

above the horizontal. If the ball lands at the
same level from which is was ki
cked, how long was it in the air?

How far did it go?

What was its final velocity?

Unit 1
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Kinematics

25

Maximum Range

The maximum range of a projectile launched over level ground occurs
when the launch angle is 45
°
.

The range is symmetric about the 45
°.

A
golfer tees off on level ground, giving the ball an initial speed of 42.0
m/s and an initial direction of 35
o

above the horizontal.

How far from the golfer does the ball land?

The next golfer hits a ball with the same initial speed, but at a greater
angle than 45
o
. The ball travels the same
horizontal distance. What was the initial direction of motion?

A cannonball is fired at an angle of 45
o

above the horizontal at an initial velocity of 77 m/s. The cannon is located
at the top of a 120 m high clif
f, and the cannonball is fired over the level plain below.

a) Draw a representation of the trajectory of the cannonball from launch until it strikes the plain below the cliff.
Label the following: A: The point where the projectile is traveling the slowest
; B: The point where the projectile has
the same speed as it does at launch; C: The point where the projectile is traveling the fastest.

b) Calculate the total time from launch until the cannonball hits the plain below the cliff.

c)
Calculate the horizontal distance that the cannonball travels before it hits the plain below the cliff.

d) Calculate the maximum height attained by the cannonball.

26

Unit 1
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Kinematics

Free Response Preparation #2

A soccer player on Krypton kicks a ball directly
toward a fence from a point 35 meters away. The initial velocity of
the ball is 25 m/s at an angle of 40
°

above the horizontal. The top of the fence is 3.0 meters above the ground.
The ball hits nothing while in flight, and, since Krypton has no atmosphere
, air resistance is nonexistent. The
acceleration due to gravity on Krypton is 12 m/s
2
.

a. Sketch the problem

b. Determine the time it takes for the ball to reach the plane of the fence.

c. Will the ball hit the fence? If so, how far below the t
op of the fence will it hit? If not, how far above the fence will
it pass?

d. Sketch the horizontal and vertical components of the ball’s velocity as functions of time until the ball reaches
the plane of the fence.

e. What is the minimum speed the soccer player must give to the ball if it is to just hit the bottom of the fence at
the same time it hits the ground?

Unit 1
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Kinematics

27

The Range Equation

One of
the important skills in physics is the skill of
deriving an equation
. At the level of this class, that will require
us to use algebra to manipulate and combine different equations to yield new equations that are useful and/or
physically insightful.

The range equation is one such
derived

equations. While you will not be asked to derive this equation on a test,
you are required to memorize the formula, since it does not appear on the formula sheet.

What is the range of a projectile that returns to th
e same vertical position from which it was launched?

28

Unit 1
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Kinematics