Physics C
Kinematics
Class Work Packet
Name:_________________________
Sample problem: From the graph, determine the
average velocity for the particle as it moves from
point A to point B.
Definition of Average Velocity
Equation:
x(m
3)
2
Definition of Average Acceleration
Equation:
1
0
A
1
2
0
3
What does the sign mean for average velocity
and acceleration?
Sample problem: A motorist drives north at 20
m/s for 20 km and then continues north at 30 m/s
for another 20 km. What is his average velocity?
B
0.
1
0.
2
0.
3
0.
5
0.
6
t(s
)
Sample problem: From the graph, determine the
average speed for the particle as it moves from
point A to point B.
x(m
3)
2
1
0
A
1
2
0
3
B
0.
1
0.
2
0.
3
0.
5
0.
6
t(s
)
Describe how you would determine the
instantaneous velocity of a particle from a graph
of its position versus time.
Sample problem: It takes the motorist one
minute to change his speed from 20 m/s to 30
m/s. What is his average acceleration?
Sample problem: From the graph, determine the
instantaneous speed and instantaneous velocity
for the particle at point B.
x(m
3)
2
1
0
A
1
2
0
3
3/6/2016
1
B
0.
1
0.
2
0.
3
0.
5
0.
6
Bertrand
t(s
)
Sample problem: Consider an object that is
dropped from rest and reaches terminal velocity
during its fall. What would the v vs t graph look
like?
Sample Problem: Estimate the net displacement
from 0 s to 4.0 s
v (m/s)
2.0
v
t
2.0
Sample problem: Consider an object that is
dropped from rest and reaches terminal velocity
during its fall. What would the x vs t graph look
like?
4.0
t (s)
Sample Problem: From the position-time graph
x
x
t
t
Sample Problem: Estimate the net change in
velocity from 0 s to 4.0 s
Draw the corresponding velocity-time graph
x
a (m/s2)
1.0
2.0
4.0
t (s)
1.0
t
Instantaneous Velocity
Definition:
Equation development:
3/6/2016
2
Bertrand
Sample problem: A particle follows the
function
Instantaneous Acceleration
Definition:
x  1.5 
4.2
 5t
t2
a) find the velocity and acceleration functions.
Equation development:
General Method for Evaluating Polynomial
Derivatives
b) find the instantaneous velocity and
acceleration at 2.0 seconds.
Write the three basic kinematic equations:
Sample problem: A particle travels from A to B
following the function x(t) = 2.0 – 4t + 3t2 –
0.2t3.
a) What are the functions for velocity and
acceleration as a function of time?
Sample problem (basic): Show how to derive
the 1st kinematic equation from the 2nd.
b) What is the instantaneous acceleration at 6
seconds?
3/6/2016
3
Bertrand
Sample problem (advanced): Given a constant
acceleration of a, derive the first two kinematic
equations.
Sample problem: A jet plane lands with a speed
of 100 m/s and can accelerate at a maximum rate
of -5.00 m/s2 as it comes to a halt.
a) What is the minimum time it needs after it
touches down before it comes to a rest?
Draw representative graphs for a particle which
is stationary.
x
v
t
b) Can this plane land at a small tropical island
airport where the runway is 0.800 km long?
a
t
t
Draw representative graphs for a particle which
has constant non-zero velocity.
x
v
a
Free Fall
Definition:
t
t
t
Acceleration during freefall:
Draw representative graphs for a particle which
has constant non-zero acceleration.
x
v
t
a
t
Sample problem: A student tosses her keys
vertically to a friend in a window 4.0 m above.
The keys are caught 1.50 seconds later.
a) With what initial velocity were the keys
tossed?
t
Sample problem: A body moving with uniform
acceleration has a velocity of 12.0 cm/s in the
positive x direction when its x coordinate is 3.0
cm. If the x coordinate 2.00 s later is -5.00 cm,
what is the magnitude of the acceleration?
b) What was the velocity of the keys just before
they were caught?
3/6/2016
4
Bertrand
Sample problem: A ball is thrown directly
downward with an initial speed of 8.00 m/s from
a height of 30.0 m. How many seconds later does
the ball strike the ground?
Sample problem: Add together the following
graphically and by component, giving the
magnitude and direction of the resultant and the
equilibrant.
– Vector A: 300 m @ 60o
– Vector B: 450 m @ 100o
– Vector C: 120 m @ -120o
Graphical method:
Graphical Addition of Vectors
In the space below, draw an addition of two
vectors A and B. Label the resultant.
Component method:
Component Addition of Vectors
1) Resolve each vector into its x- and ycomponents.
Ax = Acos
Ay = Asin
Bx = Bcos
By = Bsin
etc.
2) Add the x-components together to get Rx and
the y-components to get Ry.
3) Use the Pythagorean Theorem to get the
magnitude of the resultant.
4) Use the inverse tangent function to get the
angle.
Unit Vectors
Definition:
The unit vector pointing in the x direction is:
The unit vector pointing in the y direction is:
The unit vector pointing in the z direction is:
3/6/2016
5
Bertrand
Example: displacement of 30 meters in the +x
direction added to a displacement of 60 meters in
the –y direction added to a displacement of 40
meters in the +z direction yields a displacement
of:
Position, Velocity, and Acceleration Vectors
in Multiple Dimensions
1 Dimension
Adding Vectors Using Unit Vectors
Simply add all the i components together, all the
j components together, and all the k components
together.
2 or 3 Dimensions
Sample problem: Consider two vectors, A =
3.00 i + 7.50 j and B = -5.20 i + 2.40 j. Calculate
C where C = A + B.
Sample problem: The position of a particle is
given by r = (80 + 2t)i – 40j - 5t2k. Derive the
velocity and acceleration vectors for this particle.
Sample problem: You move 10 meters north
and 6 meters east. You then climb a 3 meter
platform, and move 1 meter west on the
platform. What is your displacement vector?
(Assume East is in the +x direction).
Sample problem: A position function has the
form r = x i + y j with x = t3 – 6 and y = 5t - 3.
a) Determine the velocity and acceleration
functions.
Sample problem: You move 10 meters north
and 6 meters east. You then climb a 3 meter
platform, and move 1 meter west on the
platform. How far are you from your starting
point?
3/6/2016
b) Determine the velocity and speed at 2
seconds.
6
Bertrand
Sample Problem: A baseball outfielder throws a
long ball. The components of the position are x =
(30 t) m and y = (10 t – 4.9t2) m
a) Write vector expressions for the ball’s
position, velocity, and acceleration as functions
of time.
Position graphs for 2-D projectiles. Assume
projectile fired over level ground.
y
x
y
x
t
t
Velocity graphs for 2-D projectiles. Assume
projectile fired over level ground.
Vy
b) Write vector expressions for the ball’s
position, velocity, and acceleration at 2.0
seconds.
Vx
t
Sample problem: A particle undergoing
constant acceleration changes from a velocity of
4i – 3j to a velocity of 5i + j in 4.0 seconds.
What is the acceleration of the particle during
this time period? What is its displacement during
this time period?
Acceleration graphs for 2-D projectiles.
ay
ax
t
3/6/2016
t
7
t
Bertrand
Sample problem: A soccer player kicks a ball at
15 m/s at an angle of 35o above the horizontal
over level ground. How far will the ball travel
until it strikes the ground?
Sample problem: Derive the range equation for
a projectile fired over level ground.
Sample problem: A cannon is fired at a 15o
angle above the horizontal from the top of a 120
m high cliff. How long will it take the
cannonball to strike the plane below the cliff?
How far from the base of the cliff will it strike?
Sample problem: Show that maximum range is
obtained for a firing angle of 45o.
Sample problem: derive the trajectory equation.
3/6/2016
8
Bertrand
Sample problem: Space Shuttle astronauts
typically experience accelerations of 1.4 g during
takeoff. What is the rotation rate, in rps, required
to give an astronaut a centripetal acceleration
equal to this in a simulator moving in a 10.0 m
circle.
Uniform Circular Motion
What is constant in uniform circular motion?
What is NOT constant in uniform circular
motion?
Draw the vectors in Uniform Circular Motion
Tangential acceleration
Definition:
Sample Problem: Given the figure at right
rotating at constant radius, find
the radial and tangential
acceleration components if  =

30o and a has a magnitude of
5.00
15.0 m/s2. What is the speed of
a
m
the particle? How is it behaving?
Centripetal acceleration
Definition:
Equation:
Sample Problem The Moon revolves around the
Earth every 27.3 days. The radius of the orbit is
382,000,000 m. What is the magnitude and
direction of the acceleration of the Moon relative
to Earth?
Sample problem: Suppose you attach a ball to a
60 cm long string and swing it in a vertical
circle. The speed of the ball is 4.30 m/s at the
highest point and 6.50 m/s at the lowest point.
Find the acceleration of the ball at the highest
and lowest points.
3/6/2016
9
Bertrand
Galileo’s Law of Transformation of Velocities
Definition:
Sample problem: A car is rounding a curve on
the interstate, slowing from 30 m/s to 22 m/s in
7.0 seconds. The radius of the curve is 30 meters.
What is the acceleration of the car?
Inertial Reference Frames
Definition:
Relative Motion
Two observers moving relative to each other at
constant velocity are both watching a third
particle. Upon which characteristics of the third
particle will the observers agree?
Sample problem: How long does it take an
automobile traveling in the left lane at 60.0km/h
to pull alongside a car traveling in the right lane
at 40.0 km/h if the cars’ front bumpers are
initially 100 m apart?
About which characteristics will they disagree?
Sample problem: Now show that although
velocity of the observers is different, the
acceleration they measure for a third particle is
the same provided vrel is constant. Begin with
vB = vA - vrel
3/6/2016
Sample problem: A pilot of an airplane notes
that the compass indicates a heading due west.
The airplane’s speed relative to the air is 150
km/h. If there is a wind of 30.0 km/h toward the
north, find the velocity of the airplane relative to
the ground.
10
Bertrand