Name: _____________________________________________
Period: _______
Table #:__________
Unit III:
Acceleration & Kinematics
© 2006 Bill Amend
Quick Review
What is Jason’s distance and displacement, in meters?
If he ran it in 45 seconds, what are his speed and velocity (in m/s)?
1
Worksheet 1
1. This is a velocity vs. time graph for an automobile on a test track. Describe how the
velocity changes with time.
a. The three notches in the graph occur where the driver changed gears. Describe
the changes in velocity and acceleration while the car was in first gear. Is the
acceleration before the gear change larger or smaller than the acceleration after
the gear change? Explain your answer
b. Use the graph above again to determine the time interval during which the
acceleration is smallest.
2. What does the slope of a tangent to the curve on a velocity vs. time graph represent?
3. What quantity is represented by the area
under the curve on a velocity-time graph?
4. Explain how you would walk to produce
both of these graphs.
2
5. Draw a velocity-time graph for each of the graphs in the figure below.
x
x
t
t
v
v
t
x
t
t
v
t
6. Using the graph below, find the acceleration of the moving object at each of the
following times.
a. During the first 5 s of travel
b. Between 5 s and 10 s
c. Between 10 s and 15 s
d. Between 20 and 25 s
3
UNIT III:
Slope & Tangent Activity
Given what you know about the motion of a cart down an incline, answer the questions that follow.
1. Using the data below, create a graph in Graphical Analysis of the position vs. Time.
Position
0.0
0.03
0.12
0.27
0.48
0.75
1.08
1.47
1.92
2.43
3.0
3.63
4.32
5.07
5.88
6.75
7.68
Time
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
2. According to your graph, what was the object doing? How does the
shape of the graph tell you this?
3. What does slope of a position vs. time graph tell you?
4. Use your slope tool and determine the slope or __________ at each
point. Record this in a new table in your notes.
5. Using your new table of data, graph Velocity vs. Time. Does this graph
support your conclusion about the motion of the graph from question 2?
How do you know?
6. What are the units for your slope? Is it constant?
7. What are the units for you y-intercept?
8. What real world value does each of the following represent? : y,m,x,b
9. What is the equation of the line for you velocity vs. time graph?
10. Using your equation in #9, what is the velocity of the object at 2 seconds?
11. Use your area tool. What is the area under the curve for the velocity vs. time graph between 0.3 s &
1.2 s? What are its units?
12. Use your table of data. What is your displacement for the time interval 0.3 s to 1.2 s?
13. What does the area under the graph of a velocity vs. time graph represent?
4
UNIT III: Worksheet 2
When evaluating problems 1 - 3, please represent the motion that would result from the rail
configuration indicated by means of a:
A) qualitative graphical representation of x vs. t
B) qualitative graphical representation of v vs. t
C) qualitative graphical representation of a vs. t
D) qualitative motion map
x
E) general mathematical expression of the
relationship between x and t
F) general mathematical expression of the
relationship between v and t
G) general mathematical expression of the
relationship between a and t
D) x
E) _____________________________
t
v
t
F) ____________________________
a
G)____________________________
t
5
2)
x
D) x
t
E) _____________________________
v
t
F) ____________________________
a
G)____________________________
t
6
3)
x
D) x
t
E) _____________________________
v
t
F) ____________________________
a
G)____________________________
t
7
When considering problems 4-5, assume that the ball does not experience any change in velocity while it is
on a horizontal portion of the rail.
Please represent the motion that would result from the rail configuration indicated by means of a:
A) qualitative graphical representation of x vs. t
B) qualitative graphical representation of v vs. t
C) qualitative graphical representation of a vs. t
D) qualitative motion map
4)
x
D) x
t
v
t
a
t
8
5)
x
D) x
t
v
t
a
t
9
Kinematic Equations Phun
For each of the graphs below, determine the slope & y-intercept as well as what they would mean.
Also determine the mathematical relationship for the line using what you calculated for the slope and
y-intercept.
Graph 1
Slope:
Units for Slope (Simplify):
Slope Represents:
Y-Intercept:
Y-Intercept Represents:
Equation for Line:
Graph 2
Slope:
Units for Slope (Simplify):
Slope Represents:
Y-Intercept:
Y-Intercept Represents:
Equation for Line:
10
Graph 3
Slope:
Units for Slope (Simplify):
Slope Represents:
Y-Intercept:
Y-Intercept Represents:
Equation for Line:
Questions
1. Look carefully at graph 1. What is happening to the velocity? What is this called? (Hint: Its
variable is “a”)
2. Now look carefully at the slope for all of the graphs. Do they have the same units? If so, what are
they?
3. How does the value for the slope from graph 1 compare to graph 2?
4. How does the value for the slope from graph 1 compare to graph 3?
5. Now look at the answers you had for the above 5 questions as well as for each graph. Using what
you have above, can you write a general equation for each of the above graphs WITHOUT numbers?
Graph
General Equation
1
2
3
11
Worksheet 3
1. Determine the final velocity of a proton that has an initial velocity of 2.35 X 105 m/s and then
is accelerated uniformly in an electric field at a rate of -1.10 X 1012 m/s2 for 1.50 X 10-7 s.
2. Marco is looking for a new sports car. He wants one with the greatest acceleration. Car A
can go from 0 m/s to 17.9 m/s in 4 s; car B can accelerate from 0 m/s to 22.4 m/s in 3.5 s;
and car C can go from 0 m/s to 26.8 m/s in 6 s. Rank the cars from greatest acceleration to
least, specifying when there are ties.
3. A supersonic jet traveling at 145 m/s experiences uniform acceleration at the rate of 23.1
m/s2 for 20 seconds.
a. What is its final velocity?
b. The speed of sound is 331 m/s. What is the plane’s speed in terms of the speed of
sound?
4. A dragster starting from rest accelerates at 49 m/s2. How fast is he going when he has
traveled 325m?
12
5. A car travels at 12 m/s and coasts up a hill with uniform acceleration of -1.6 m/s2.
a. What is its displacement after 6 s?
b. What is its displacement after 9 s?
6. A racecar can be slowed with a constant acceleration of -11 m/s2.
a. If the car is going 55 m/s, how far will it travel before it stops?
b. How many meters will it take to stop a car going twice as fast?
7. Determine the displacement of a plane that experiences uniform acceleration from 66 m/s to
88 m/s in 12 seconds.
8. How far does a plane travel in 50 s when its velocity is changing from 145 m/s to 75 m/s at a
uniform rate of acceleration?
13
Worksheet 4
1.
A skateboarder is moving at a constant velocity of 1.75 m/s when she starts up an incline
that causes her to slow down at a constant acceleration of -0.2 m/s2. How much time
passes from when she begins to slow down to when she begins to roll back down the
incline?
2.
Lightning McQueen travels on a race track at 44 m/s and slows at a constant rate to a
velocity of 22 m/s in 11 s. How far does he move over this time?
3.
A car accelerates at a constant rate from 15 m/s to 25 m/s while it travels 125 meters.
How long does it take to achieve this speed?
4.
Lance pedals with a constant acceleration to reach a velocity of 75 m/s in 4.5 s. During
the period of acceleration, the bikes displacement is 19 m. What was the initial velocity
of the bike?
5.
Phil A Buster runs at a velocity of 4.5 m/s for 15 minutes. When going up an
increasingly steep hill, he slows at a constant rate of 0.05 m/s2 for 90 s and then comes
to a stop. How far did he run?
14
6.
Time
(s)
0
1
2
3
4
5
6
Robin Banks is learning to ride a bike without training wheels. Her father pushes her
with a constant acceleration of 0.5 m/s2 for 6 s, and then Robin pedals at 3 m/s for 6 sec
before falling. What is her displacement? (construct a velocity-time graph and find the
area under the graphed line)
Velocity
(m/s)
0
7.
You start your bicycle ride at the top of a hill. You coast down the hill at a constant
acceleration of 2 m/s2. When you get to the bottom of the hill, you are moving at 18 m/s,
and you pedal to maintain that speed. If you continue at this speed for 1 min, how far
have you traveled since the top of the hill?
8.
Russ T. Nale is training for a 5 km race. She starts out her training moving at a constant
rate of 4.3 m/s for 19 min. Then she accelerates at a constant rate until she finishes the
race, 19.4 seconds later. What is her acceleration during the last portion of the training
run?
15
Worksheet 5
1. A woman driving at a constant speed of 23 m/s sees a deer on the road ahead and applies
the brakes when she is 210 m from the deer. If the deer does not move and the car stops
right before it hits the deer, what is the acceleration applied by the brakes?
2. If you were given initial and final velocities and the constant acceleration of an object, and
you were asked to find the displacement, what equation would you use?
3. An in-line skater first accelerates from 0.0 m/s to 5.0 m/s in 4.5 s, and then continues at this
constant speed for another 4.5 s. What is her total displacement traveled?
4. A plane travels a distance of 5.0 X 102 m while being accelerated uniformly from rest at
5.0 m/s2. What final velocity does it attain?
16
5. An airplane accelerated uniformly from rest at a rate of 5.0 m/s2 for 15 s. What final velocity
does it attain?
6. An airplane starts at rest and accelerates uniformly at a rate of 3.0 m/s2 for 30 s before
leaving the ground.
a. How far did it move?
b. How fast was it going when it took off?
7. A sprinter walks up to the starting block at a constant speed and positions herself for the
start of the race. She waits until she hears the starting pistol go off, and then accelerates
rapidly until she reaches a constant velocity. She maintains this velocity until she crosses
the finish line, and then she slows down to a walk, taking more time to slow down than she
did at the beginning of the race. Sketch a velocity- time and position- time graph to
represent her motion. Draw them one above the other on the same time scale. Indicate on
your positions-time graph where the starting blocks are and where the finish line is.
17
Unit 3 Worksheet 6
1. Use the graph to answer the following questions…Jason and his sister, Tara, are riding bicycles.
Jason tries to catch up to Tara, who has a 10 sec head start
V
e
l
8
Jason
6
Tara
4
(m/s)
2
0
10
20
30
Time (Sec)
40
a. What is Jason’s Acceleration?
b. What is Tara’s acceleration?
c. At what time do they have the same velocity?
2. A dragster accelerates for 4 seconds at a rate of 5 m/s/s. It then travels at a constant speed for 2.5
seconds. A parachute opens, stopping the vehicle at a constant rate in 2 seconds. Plot the v-t graph
(Quantitative) representing the motion of the dragster during the entire motion.
3. A car traveling at 21 m/s misses the turnoff on the road and collides into the safety guard rail. The car
comes to a stop in 0.55 seconds.
a. What is the average acceleration of the car?
b. If the safety rail consisted of a section of rigid rail, the car would stop in 0.15 seconds. What
would be the acceleration in this case?
4. On the way to the school, Willie Makit realizes that he left his physics homework at home. His car
was initially heading north at 36 m/s. It takes him 35 seconds to turn his car around and head south at
25 m/s. If north is designated to be the positive direction, what is the average acceleration of the car
during the 35 second interval?
18
5. A cheetah can reach a top speed of 27.8 m/s in 5.2 seconds. What is the cheetah’s average
acceleration?
6. After being launched, a rocket attains a speed of 122 m/s before the fuel in the motor is completely
used. The acceleration of the rocket is 32.2 m/s/s, how much time does it take for the fuel to be
completely consumed?
7. An object in free-fall has an acceleration of 9.8 m/s/s (with no air resistance). What will be the speed
of an object that is dropped from a tall cliff 3.5 seconds after it has been released?
19
UNIT III: Worksheet 7
While cruising along a dark stretch of highway with the cruise control set at 25 m/s (≈55 mph), you see, at
the fringes of your headlights, that a bridge has been washed out. You apply the brakes and come to a stop
in 4.0s. Assume the clock starts the instant you hit the brakes.
1. Construct a motion map that represents the motion described above, including position, velocity, and
acceleration. Clearly demonstrate how you can determine the direction (sign) of the acceleration from the
motion map representation.
2. Construct qualitative graphical representations of the situation described above to illustrate:
a. x vs. t
b. v vs. t
c. a vs. t
20
3. Construct a quantitatively accurate v vs t graph to
describe the situation.
4. On the v vs t graph at right, graphically represent the
car’s displacement during braking.
5. Utilizing the graphical representation, determine
how far the car traveled during braking. (Please explain
your problem solving method.)
6. In order to draw the a vs t graph, you need to determine
the car’s acceleration. Please do this, then sketch a
quantitatively accurate a vs t graph
7. Using the equation you developed for displacement of an accelerating object determine how far the car
traveled during braking. (Please show your work.)
8. Compare your answers to 5 and 7.
21
UNIT III: Worksheet 8
This time, while cruising along a dark stretch of highway at 30 m/s (≈65 mph), you see, at the fringes of
your headlights, some roadkill on the highway. It takes you 0.5 s to react, then you apply the brakes and
come to a stop 3.5s later. Assume the clock starts the instant you see the hazard.
1. Construct a motion map that represents the motion described above, including position, velocity, and
acceleration. Hint: make the dots at 0.5s intervals.
2. Construct a quantitatively accurate v vs t graph to
describe the situation.
3. On the v vs t graph at right, graphically represent the
car’s displacement during this incident.
4. Utilizing the graphical representation, determine
how far the car traveled during this incident. (Please
explain your problem solving method.)
5. In order to draw the a vs t graph, you need to determine
the car’s acceleration once the brakes were applied.
Please do this, then sketch a quantitatively accurate a vs
t graph
22
6. Two kinds of motion occur in this case. For the first 0.5s, the car is traveling at constant
velocity. For the remainder of the time, the car has an initial velocity and a uniform acceleration. Using
the appropriate mathematical representation for each phase of the motion, determine how far the car
traveled from the instant you noticed the hazard until you came to a stop. As always, show work and
include units.
7. Compare your answers to 4 and 6.
23
UNIT III: Worksheet 9
1.
x (m)
25
0
5
t (s)
a. Describe in words the motion of the object from 0 - 6.0 s.
b. Construct a qualitative motion map to describe the motion of the object depicted in the graph above.
c. What is the instantaneous velocity of the object at the following times?
i.
t = 1.0 s
ii.
t = 3.0 s
d. What is the simple average of these two velocities?
What is the average velocity for the entire interval?
Why are these two values different? Which is best to describe the motion of the object?
24
v (m/s)
e. Graphically represent the relationship between velocity and time for the object described above.
0
t (s)
5
f. From your velocity vs. time graph determine the total displacement of the object.
2. The graph below represents the motion of an object.
x
B
A
C
t
D
E
G
F
a. At what point(s) on the graph above is the object moving most slowly? (How do you know?)
b. Over what intervals on the graph above is the object speeding up? (How do you know?)
c. Over what intervals on the graph above is the object slowing down? (How do you know?)
d. At what point(s) on the graph above is the object changing direction? (How do you know?)
25
3. A stunt car driver testing the use of air bags drives a car at a constant speed of 25 m/s for a total of 100m.
He applies his brakes and accelerates uniformly to a stop just as he reaches a wall 50 m away.
a.
Sketch qualitative position vs. time and velocity vs time graphs.
b. How long does it take for the car to travel the first 100.m?
c. Remember that the area under a velocity vs time graph equals the displacement of the car. How long
must the brakes be applied for the car to come to a stop in 50 m?
d.
Now that you know the total time of travel, sketch a quantitative velocity vs time graph.
e. What acceleration is provided by the brakes? How do you know?
26
UNIT III: Worksheet 10
+
v (m/s)
1. A poorly tuned Yugo can accelerate from rest to 28 m/s in 20 s.
a) What is the average acceleration of the car?
b) What distance does it travel in this time?
t (s)
-
+
v (m/s)
2. At t = 0 a car has a speed of 30 m/s. At t = 6 s, its speed is 14 m/s.
What is its average acceleration during this time interval?
t (s)
-
+
v (m/s)
3. A bear spies some honey and takes off from rest, accelerating at a
rate of 2.0 m/s2. If the honey is 16 m away, how fast will his snout
be going at the moment of ecstasy?
t (s)
-
+
v (m/s)
4. A bus moving at 20 m/s (t = 0) slows at a rate of 4 m/s each second.
a) How long does it take the bus to stop?
b) How far does it travel while braking?
t (s)
-
27
+
v (m/s)
5. A physics student skis down a hill, accelerating at a constant
2.0 m/s2. If it takes her 15 s to reach the bottom, what is the length
of the slope?
t (s)
-
+
v (m/s)
6. A dog runs down his driveway with an initial speed of 5 m/s for 8 s,
then uniformly increases his speed to 10 m/s in 5 s.
a) What was his acceleration during the 2nd part of the motion?
b) How long is the driveway?
t (s)
-
v (m/s)
7. A mountain goat starts a rock slide and the rocks crash down the slope
100 m. If the rocks reach the bottom in 5 s, what is their acceleration? +
t (s)
-
v (m/s)
8. A car whose initial speed is 30 m/s slows uniformly to 10 m/s in 5 s.
+
a) Determine the acceleration of the car.
b) Determine the distance it travels in the 3rd second (t = 2s to t = 3s).
t (s)
-
28
Worksheet 11: Stacks of kinematics curves
Given the following position vs time graphs, sketch the corresponding velocity vs time and
acceleration vs time graphs.
29
For the following velocity vs time graphs, draw the corresponding position vs time and acceleration vs
time graphs.
30
UNIT III: Review
Use the graph below to answer questions #1-4 that follow:
1. Give a written description to describe the motion of this object.
2. Draw the motion map for the object. Include velocity and acceleration vectors.
3. Explain how you could determine the instantaneous velocity of the object at t = 2 s.
4. Assume the initial velocity was 50 m/s; determine the acceleration of the object.
5. A Pontiac Trans-Am, initially at rest, accelerates at a constant rate of 4.0 m/s2 for 6 s. How fast will the
car be traveling at t = 6 s?
6. A tailback initially running at a velocity of 5.0 m/s becomes very tired and slows down at a uniform rate
of 0.25 m/s2. How fast will he be running after going an additional 10 meters?
31
7. For each of the position vs time graphs shown below, draw the corresponding
v vs t, a vs t , and motion map.
8.
Using the graph below, compare the kinematic behavior of the two objects.
Comparison:
is A > B, A < B, or A = B,
How do you know?
a. Displacement at 3 s
b. Average velocity from 0 - 3 s
c. Instantaneous velocity at 3 s
32
Unit III: Lab Review
9. Speeding up, moving in the positive direction
a. Observe the motion of the cart starting from rest and rolling down the incline without using the
motion detector.
b. Draw a motion map including both velocity and acceleration vectors.
c. Is the velocity positive or negative?
e. Predict the graphs describing the motion.
d. Is the acceleration positive or negative?
f. Record the graphs as displayed by the
motion detector.
g. On the observed graphs, describe the slope as
a) constant, increasing or decreasing
b) positive or negative
c) state what the slope represents
Note: You may have to divide your graph into segments.
33
10. Slowing down, moving in the positive direction
a. Observe the motion of the cart slowing after an initial push without using the motion detector.
Answer the following questions for the cart while coasting.
b. Draw a motion map including both velocity and acceleration vectors.
c. Is the velocity positive or negative?
e. Predict the graphs describing the motion.
d. Is the acceleration positive or negative?
f. Record the graphs as displayed by the
motion detector.
g. On the observed graphs, describe the slope as
a) constant, increasing or decreasing
b) positive or negative
c) state what the slope represents
Note: You may have to divide your graph into segments.
34
11. Speeding up, moving in the negative direction
a. Observe the motion of the cart starting from rest and rolling down the incline without using the
motion detector.
b. Draw a motion map including both velocity and acceleration vectors.
c. Is the velocity positive or negative?
e. Predict the graphs describing the motion.
d. Is the acceleration positive or negative?
f. Record the graphs as displayed by the
motion detector.
g. On the observed graphs, describe the slope as
a) constant, increasing or decreasing
b) positive or negative
c) state what the slope represents
Note: You may have to divide your graph into segments.
35
12. Slowing down, moving in the negative direction
a. Observe the motion of the cart slowing after an initial push without using the motion detector.
Answer the following questions for the cart while coasting.
b. Draw a motion map including both velocity and acceleration vectors.
c. Is the velocity positive or negative?
e. Predict the graphs describing the motion.
d. Is the acceleration positive or negative?
f. Record the graphs as displayed by the
motion detector.
g. On the observed graphs, describe the slope as
a) constant, increasing or decreasing
b) positive or negative
c) state what the slope represents
Note: You may have to divide your graph into segments.
36
13. Up and down the ramp
a. Observe the motion of the cart after an initial push without using the motion detector. Answer
the following questions for the cart while coasting.
b. Draw a motion map including both velocity and acceleration vectors.
c. Is the velocity positive or negative?
Does the direction of the velocity change?
e. Predict the graphs describing the motion.
d. Is the acceleration positive or negative?
Does the direction of the acceleration change?
f. Record the graphs as displayed by the
motion detector.
g. On the observed graphs, describe the slope as
a) constant, increasing or decreasing
b) positive or negative
c) state what the slope represents
Note: You may have to divide your graph into segments.
37
14. Up and down the ramp with a different zero position
a. Observe the motion of the cart after an initial push without using the motion detector. Answer
the following questions for the cart while coasting.
b. Draw a motion map including both velocity and acceleration vectors.
c. Is the velocity positive or negative?
Does the direction of the velocity change?
Is position A positive or negative?
e. Predict the graphs describing the motion.
Label points A and B on your x-t graph.
d. Is the acceleration positive or negative?
Does the direction of the acceleration change?
Is position B positive or negative?
f. Record the graphs as displayed by the
motion detector.
g. On the observed graphs, describe the slope as
a) constant, increasing or decreasing
b) positive or negative
c) state what the slope represents
Note: You may have to divide your graph into segments.
38
Calculation Answers for Unit 3 Packet
Since you have the answers to these questions, your teacher will be looking for the work as to how you got to the
answers to receive credit. Be sure to ALWAYS show ALL of your work for ANY calculations!!
Worksheet 1
6a. 6 m/s
6b. 30 m/s
6c. -2 m/s
6d. -4 m/s
Worksheet 5
1. -1.26 m/s2
3. 33.75 m
4. 70.71 m/s
5. 76 m/s
6a. 1350 m
6b. 90 m/s
Worksheet 3
1. 7,000 m/s
2. Car A: 4.475 m/s2 Car B: 6.4
m/s2 Car C: 4.46 m/s2
3a. 607 m/s
3b. 1.8x faster
4.178.46 m/s
5a. 43.2 m
5b. 7.2 m
6a. 137.5 m
6b. 550 m
7. 942 m
8. 5500 m
Worksheet 6
1. 8.8 sec
2. 363 m
3.6.3 sec
4. 0.944 m/s
5. 4252.5 m
7. 1161 m
8. 0.08 m/s2
1a. 0.4 m/s2
1b. 0.2 m/s2
3a. -38.2 m/s2
3b. -140 m/s2
4. -1.7 m/s2
5. 5.3 m/s2
6. 3.8 sec
7. 34.3 m/s
5. 50 m
7. 50 m
Worksheet 8
4. 67.5 m
5. -8.6 m/s2
6. 67.5 m
Worksheet 4
Worksheet 7
Worksheet 9
Worksheet 10
1a. 1.4 m/s2
1b. 280 m
2. -2.7 m/s2
3. 8 m/s
4a. 5 sec
4b. 50 m
5. 225 m
6a. 1 m/s2
6b. 77.5 m
7. 8 m/s2
8a. -4 m/s2
8b. 20 m
1c. 0 m/s ; 7.5 m/s
1d. 3.75 m/s ; 5 m/s
1f. 30 m
3b. 4 sec
3c. 4 sec
3e. -6.25 m/s2
Review
3. -6 m/s
4. -9.2 m/s2
5. 24 m/s
6. 4.5 m/s
39