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KINEMATICS www.graceimson.weebly.com Instructor: Grace Imson

Kinematics: the branch of mechanics that describes the

motion of objects without necessarily discussing what

causes the motion.

Distance

Definition:

SI Unit:

Displacement (x)

Definition:

Equation:

SI Unit:

Question: Does the odometer in your car measure

distance or displacement?

Can you think of a circumstance when it would measure

both distance and displacement?

Practice Problem: Two tennis players approach the net

to congratulate one another after a game.

a) Find the distance and displacement of player A.

b) Repeat for player B.

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 = |x|

b) d < |x|

c) d > |x|

d) d > |x|

e) d < |x|

Practice Problem: A particle moves from x = 1.0 meter

to x = -1.0 meter.

a) What is the distance d traveled by the particle?

b) What is the displacement of the particle?

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?

Average Speed

Definition:

Equation:

SI unit:

A B 5 m 2 m

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Average Velocity

Definition:

Equation:

SI unit:

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.

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?

Graphical Problem

What physical feature of the graph gives the constant

velocity from A to B?

Graphical Problem: Determine the average velocity

from the graph.

Graphical Problem: Determine the average velocity

between 1 and 4 seconds.

t

x A

B

x (m)



Instantaneous Velocity

Definition:

Practice Problem: Determine the instantaneous

velocity at 1.0 second.

Acceleration

Definition:

What does the sign of the acceleration signify?

What types of acceleration are there?

Questions

If acceleration is zero, what does this mean about the

motion of an object?

Is it possible for a racecar circling a track to have zero

acceleration?

Uniform (Constant) Acceleration

Equation:

SI unit:

Acceleration in 1-D motion has a sign!

If the sign of the velocity and the sign of the

acceleration is the same, what happens?

If the sign of the velocity and the sign of the

acceleration are different, what happens?

Practice Problem: A 747 airliner reaches its takeoff

speed of 180 mph in 30 seconds. What is its average

acceleration?

Practice Problem: A horse is running with an initial

velocity of 11 m/s, and begins to accelerate at –1.81

m/s2. How long does it take the horse to stop?

Graphical Problem

What physical feature of the graph gives the

acceleration?

t

v A

B



Practice Problem: Determine the acceleration from the

graph.

Determine the displacement of the object from 0 to 4.0

seconds (using the graph above)

How would you describe the motion of this particle?

Describe the motion

Draw Graphs for Stationary Particles

Draw Graphs for Constant Non-zero Velocity

Draw Graphs for Constant Non-zero Acceleration

Kinematic Equations

Equation 1:

Equation 2:

Equation 3:

Practice Problem: What must a particular Olympic

sprinter’s acceleration be if he is able to attain his

maximum speed in ½ of a second?

t

x

t

v

t

a

t

x

t

v

t

a

t

x

t

v

t

a



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?

Practice Problem: On a ride called the Detonator at

Worlds of Fun in Kansas City, passengers accelerate

straight downward from 0 to 20 m/s in 1.0 second.

a) What is the average acceleration of the passengers on

this ride?

b) How fast would they be going if they accelerated for

an additional second at this rate?

c) Sketch approximate x-vs-t, v-vs-t and a-vs-t graphs

for this ride.

Practice Problem: Air bags are designed to deploy in

10 ms. Estimate the acceleration of the front surface of

the bag as it expands. Express your answer in terms of

the acceleration of gravity g.

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/s2.

a) How far do you travel before stopping?

b) When you have traveled only half the stopping

distance, what is your speed?

c) How long does it take you to stop?

d) Draw x vs t, v vs t, and a vs t graphs for this.

Free Fall

Definition:

Acceleration due to Gravity:



Practice Problem: You drop a ball from rest off a 120 m high cliff.

Assuming air resistance is negligible,

a) how long is the ball in the air?

b) what is the ball’s speed and velocity when it strikes

the ground at the base of the cliff?

c) sketch approximate x-vs-t, v-vs-t, a-vs-t graphs for

this situation.

Practice Problem: You throw a ball straight upward into the air with a

velocity of 20.0 m/s, and you catch the ball some time

later.

a) How long is the ball in the air?

b) How high does the ball go?

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

d) Sketch approximate x-vs-t, v-vs-t, a-vs-t graphs for

this situation.

Symmetry in Free Fall

When something is thrown straight upward under

the influence of gravity, and then returns to the thrower,

this is very symmetric.

The object spends half its time traveling up; half

traveling down.

Velocity when it returns to the ground is the

opposite of the velocity it was thrown upward with.

Acceleration is 9.8 m/s2

and directed DOWN the

entire time the object is in the air!

Homework Problem: Below is some data for a car in

the Pinewood Derby. Using these data, work the

following problem:

Pinewood Derby

5.04.03.02.01.00t(s)

57.536.820.79.22.30x(m)

On your graph paper, do the following.a) Draw a position vs time graph for the car.b) Draw tangent lines at three different points on the curve to determine the instantaneous velocity at all three points.c) On a separate graph, draw a velocity vs time graph using the instantaneous velocities you obtained in the step above.d)From your velocity vs time graph, determine the acceleration of the car.



Homework problem:

Work this problem on a separate sheet of

graph paper to turn in.

A world-class runner can complete a 100 m

dash in about 10 s. Past studies have shown

that runners in such a race accelerate

uniformly for a time tu and then run at constant

speed for the remainder of the race. A world-

class runner is visiting your physics class. You

are to develop a procedure that will allow you

to determine the uniform acceleration au and

an approximate value of tu for the runner in a

100 m dash. By necessity your experiment will

be done on a straight track and include your

whole class of eleven students.

(a) Indicate which of the following pieces

of equipment, other than the runner and

the track, that your class will need to

do the experiment. List the equipment

you use in your experiment on your

graph paper.

_____ Stopwatches

_____Tape measures

_____Rulers

_____Masking tape

_____Metersticks

_____Starter’s pistol

_____String

_____Chalk

(b) Outline the procedure that you would

use to determine au and tu , including a

labeled diagram of the experimental

setup. Use symbols to identify

carefully what measurements you

would make and include in your

procedure how you would use each

piece of the equipment you checked in

part (a).

(c) Outline the process of data analysis,

including how you will identify the

portion of the race that has uniform

acceleration, and how you would

calculate the uniform acceleration.

NOTE: TO EARN FULL CREDIT FOR

THIS PART, YOU MUST COME UP

WITH A GRAPHICAL SOLUTION!



Homework problem 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 vo. Derive an expression for the maximum

height ymax reached by the ball.

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 vo 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 vo from

the foot of the cliff directly beneath ball A.

Derive an expression for how high above the

ground they will collide.

B

h

vo

A

h



2-Dimensional Motion

Definition:

Examples:

Solving 2-D Problems

Resolve all vectors into components.

Work the problem as two one-dimensional

problems.

Re-combine the results for the two components

at the end of the problem.

Problem-1

You run in a straight line at a speed of 5.0 m/s

in a direction that is 40o south of west. a) How far west have you traveled in 2.5

minutes?

b) How far south have you traveled in 2.5

minutes?

Problem-2

A roller coaster rolls down a 20o incline with an

acceleration of 5.0 m/s2. a) How far horizontally has the coaster

traveled in 10 seconds?

b) How far vertically has the coaster

traveled in 10 seconds?

Sample Problem

A particle passes through the origin with a

speed of 6.2 m/s in the positive y direction. If

the particle accelerates in the negative x

direction at 4.4 m/s2 a) What are the x and y positions at 5.0

seconds?

b) What are the x and y components of

velocity at this time?



Projectile Motion

Something is fired, thrown, shot, or hurled near

the earth’s surface.

Horizontal velocity is __________

Vertical velocity is ____________

Air resistance is _____________

1-Dimensional Projectile

Definition:

Examples:

You calculate vertical motion only.

The motion has no horizontal component.

2-Dimensional Projectile

Definition:

Examples:

You calculate vertical and horizontal motion.

Horizontal Component of Velocity

Characteristics:

Equation:

Vertical Component of Velocity

Characteristics:

Equations:

Launch angle

Definition:

Why is the launch angle important?

Zero Launch angle

What does a zero launch angle imply?

Exit Problem (Turn-In before you leave the class)

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 water when it hits the bottom?

Assume the water is in freefall as it drops.



Sample Problem

An astronaut on the planet Zircon tosses a rock

horizontally with a speed of 6.75 m/s. The rock

falls a distance of 1.20 m and lands a horizontal

distance of 8.95 m from the astronaut. What is

the acceleration due to gravity on Zircon?

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. a) How far were you from the second

baseman?

b) What is the distance of the vertical drop?

Resolving the velocity

Use speed and the launch angle to find

horizontal and vertical velocity components.

Then proceed to work problems just like you did

with the zero launch angle problems.

Sample problem

A soccer ball is kicked with a speed of 9.50 m/s

at an angle of 25o above the horizontal. If the

ball lands at the same level from which is was

kicked, how long was it in the air?

Sample problem

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; snowball B is

thrown in a direction 25o 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.

Trajectory of a 2-D Projectile launched over level

ground Sketch

Characteristics:

Range of a 2-D Projectile launched over level

ground Definition:

Range of a 2-D Projectile launched over level

ground Definition

Acceleration of a projectile launched over level ground

Notes:

Velocity of a projectile launched over level ground

Notes:

Time of flight for a projectile launched over level ground

Notes:

Position graphs for 2-D projectiles

Velocity graphs for 2-D projectiles

Acceleration graphs for 2-D projectiles

Sample problem

A golfer tees off on level ground, giving the ball

an initial speed of 42.0 m/s and an initial

direction of 35o above the horizontal. a) How far from the golfer does the ball land?

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



Free Response #1 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 cliff, 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.



Free Response #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 40o

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/s2.

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 top 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. Draw and label your axes!

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?


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