Post on 23-Jan-2022
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
KINEMATICS www.graceimson.weebly.com Instructor: Grace Imson
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)
KINEMATICS www.graceimson.weebly.com Instructor: Grace Imson
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
KINEMATICS www.graceimson.weebly.com Instructor: Grace Imson
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
KINEMATICS www.graceimson.weebly.com Instructor: Grace Imson
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:
KINEMATICS www.graceimson.weebly.com Instructor: Grace Imson
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.
KINEMATICS www.graceimson.weebly.com Instructor: Grace Imson
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!
KINEMATICS www.graceimson.weebly.com Instructor: Grace Imson
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
KINEMATICS www.graceimson.weebly.com Instructor: Grace Imson
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?
KINEMATICS www.graceimson.weebly.com Instructor: Grace Imson
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.
KINEMATICS www.graceimson.weebly.com Instructor: Grace Imson
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.
KINEMATICS www.graceimson.weebly.com Instructor: Grace Imson
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?
KINEMATICS www.graceimson.weebly.com Instructor: Grace Imson
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.
KINEMATICS www.graceimson.weebly.com Instructor: Grace Imson
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?