Tutorial 2 Homework
Name
Graphing and data analysis
SOLUTION
Tutorial section
In the tutorial about velocity and acceleration graphs last week, most of you didn’t finish the last
set of questions about a cart going up and down a ramp. The first two pages below cover that same
tutorial material. If you didn’t complete all of the tutorial questions do them below, working with a
group, if possible. DO NOT hand in these first two (tutorial) pages with this homework .
I.
Up and down a ramp
In all these questions, just think about the cart’s motion after you stop
pushing it up the ramp but before you catch it on the way down.
A. Velocity graph review.
detector
velocity
1. This graph shows the cart’s velocity vs. time. On
those same axes, graph the cart’s speed vs. time.
Speed is how fast something is moving, independent
of direction.
time
[Speed shown in gray]
2. What would you guess is a common mistake students
make on the velocity vs. time graph for this and
similar situations? Explain.
peak
Graphing speed instead of velocity, i.e., not taking into account the direction of motion.
B. Acceleration on the way down
1. Consider the cart as it rolls down the ramp after reaching its peak. Give a reason why a smart
student might think the cart’s acceleration during that segment of the motion is positive. Briefly
summarize the reasoning.
Since it’s speeding up, the acceleration is positive. If it were slowing down, the acceleration would
be negative.
2. Now give a reason why a smart student might think the cart’s acceleration during that segment of
the motion is negative. Briefly summarize the reasoning.
The velocity is still going “down,” as the velocity graph shows. Also, motion towards the detector
“flips” the sign of the velocity, so it makes sense that it would also “flip” the sign of the
acceleration.
3. Below, sketch two different predictions for the cart’s acceleration vs. time. On "Prediction
graph 1," assume the cart’s acceleration as it rolls down the ramp is positive (argument 1 above).
On "Prediction graph 2," assume the acceleration is negative during that segment (argument 2).
Prediction graph 1
Prediction graph 2
acceleration
acceleration
time
peak
© University of Maryland Physics Education Research Group, Fall 2004.
time
peak
HW2-1
Tutorial 2 Homework
Name
Graphing and data analysis
Tutorial section
C. When performed with good equipment, the acceleration graph comes out as shown here. Two
students disagree about what they should do next to continue learning about acceleration.
VENUS: Look, now we know which argument to
acceleration
believe about whether the acceleration is
positive or negative when the cart rolls
down the ramp. Physics is an experimental
science! Since we know the right answer
and the reasoning behind it, we’re ready to
move on.
time
peak
SERENA: But we also came up with a sensible
argument that the acceleration would be positive, not negative. To understand
acceleration, I think we also need to understand what’s wrong with that argument.
Which student do you agree with more in this case? Explain.
Good arguments can be made either way. Here’s one possible compromise. To take Serena’s idea
into account, Venus could say that, in this particular case, the explanation of why acceleration is
negative coming down the ramp already contains, “built into it,” a fairly clear explanation of why the
acceleration isn’t positive.
D. How to decide?
1. Consider the contradictory arguments from part B about whether the cart’s acceleration is
positive or negative as it rolls down the ramp. Was it possible to decide for sure, without doing
the experiment, which argument was correct? Explain.
You can decide the issue by checking for coherence between the velocity and acceleration graphs.
The velocity graph is negatively sloped the whole time, including when the cart rolls back down the
ramp. So, the acceleration is negative the whole time.
2. Let’s see if the president-for-life of mistake-avoidance strategies can help us here. Look at the
velocity graph on the previous page.. By checking for coherence between your velocity graph
predictions and your acceleration graph, could you decide whether the acceleration is positive or
negative as the cart rolls down the ramp? Explain. (If your above answers addressed this issue,
skip this question.)
See above answer.
3. We just saw that checking for coherence can help you decide between sensible competing
arguments. Can that game also help you understand why the other argument is wrong? See if
you can use the connection between velocity and acceleration graphs to explain, in a common
sense way, why the cart’s acceleration is negative even though it’s gaining speed (as it rolls down
the ramp)? Hint: Is the cart’s velocity (not speed) trending “up” or “down?”
Intuitively speaking, acceleration is positive when the velocity goes up and negative when the
velocity goes down. As the velocity vs. time graph for this experiment shows, the velocity is still
“going down” when the cart rolls down the ramp. In general, when something negative gets more and
more negative, it’s “going down.” For instance, if your bank balance is negative (meaning you owe the
bank money!) and it keeps getting more negative, your net worth is going down, not going up!
9
© University of Maryland Physics Education Research Group, Fall 2004.
HW2-2
Tutorial 2 Homework
Name
Graphing and data analysis
II.
Tutorial section
Rolling ball
A ball, released from rest, rolls down ramp A, then along the floor, then up
ramp B, as drawn here.
A
B
A. Sketch the ball’s velocity vs. time and acceleration vs. time, from the
moment it’s released until the moment it reaches its peak on the second ramp. Label your graphs in
such a way that we can tell when the ball is on ramp A, when it’s on the floor, and when it’s on ramp
B. Remember to end your graphs at the moment the ball reaches its highest point on ramp B.
acceleration
velocity
time
time
bottom
of A
bottom
of B
peak
of B
bottom
of A
B. Now sketch distance vs. time. (“Distance” here means the distance the
ball has rolled along the track from its starting place.) Explain your
reasoning in words.
bottom
of B
distance
On ramp A, the ball gains speed, which means the slope of the position
graph increases; the ball covers distance at an increasing rate. On the
floor, the ball “keeps” the high speed it attained and rolls at constant
velocity, indicated on the position graph by a constant slope (straight line).
On ramp B, the ball slows down but doesn’t reverse direction (until after
the graph ends). So, the position graph keeps going up, though at a
decreasing rate (slope).
C. Bill got his position and acceleration graphs right but drew the incorrect
velocity vs. time graph shown here.
peak
of B
bottom
of A
bottom
of B
time
velocity
1. Why do you think Bill make this mistake? What advice would you
give him to help him avoid that mistake in the future?
He seemed to be thinking that something about the motion “stops” on
the floor. That’s true about the acceleration, not the velocity. When he
wants to indicate that something “stops,” Bill should think carefully
whether the object stops moving (zero velocity) or stops speeding up or slowing down (zero
acceleration).
time
2. After drawing his velocity graph, how could Bill have realized that it’s wrong? (If you already
answered this in question 1, just say so.)
Checking for coherence between the velocity and acceleration graphs would not catch the mistake
here. Even though Bill’s velocity graph is wrong, it “fits together” with the acceleration graph
because the slope of his velocity graph in each segment does indeed equal the ball’s acceleration.
(Make sure you understand why.) However, Checking for Coherence between velocity and position
would catch the mistake, because Bill’s velocity graph doesn’t fit together with the position graph.
9
© University of Maryland Physics Education Research Group, Fall 2004.
HW2-3
Tutorial 2 Homework
Name
Graphing and data analysis
III.
Tutorial section
Velocity graph
Consider the velocity vs. time graph shown here, focusing on the issue of whether
the object turns around (reverses direction).
velocity
time
A. Why might a smart student say the object reverses direction?
The graph switches from trending up to trending down—a switch in direction.
B. Why might a smart student say the object doesn’t reverse direction?
The velocity stays positive the whole time, meaning the object always moves away from the
detector.
C. Does the object reverse direction? As part of your explanation, explain why the other argument is
incorrect, and give advice to someone who made that mistake that can help them understand why they
made the mistake and how they can avoid it in the future.
Argument B is correct. The object “switches” from speeding up to slowing down, but it keeps
moving in the same direction. If this were a position graph, it would indicate a reversal of direction.
When someone sees a slope “reversal” on a graph, he or she needs to think carefully about whether
the reversal is from speeding up to slowing down (as is the case on a velocity graph) or a reversal in
the direction of motion (as is the case on a position graph).
position
D. Sketch a graph of the object’s position vs. time.
time
IV.
Bouncing ball
A hard rubber ball is dropped from rest. It falls to the concrete
floor and bounces back up almost to its initial height. A motion
detector is mounted on the ceiling directly above the ball, facing
down. So, in this example, the positive direction—the awayfrom-the-detector direction—is downward.
A. On these graphs, sketch the ball’s position, velocity and
acceleration vs. time. Treat the bounce as a sudden event
that happens in a single instant of time. (Below, you’ll think
about what’s going on during the bounce.) Because you’re
dealing with all three kinds of graphs at once, think carefully
and use the strategies you’ve learned for avoiding and
catching mistakes.
Position
time
bounce
Velocity
B. Two students discuss the acceleration during the bounce,
i.e., during the brief time the ball is in contact with the floor.
time
SMITHA: The ball’s acceleration must be large during the
bounce, larger than when it’s just falling or rising.
Because during the bounce, the ball’s velocity
changes a lot, very suddenly, a big change in
velocity over a very small time. So, the
acceleration — the rate of change of velocity — is
big.
bounce
Acceleration
We’re leaving out
time
what happens
during the bounce
bounce
© University of Maryland Physics Education Research Group, Fall 2004.
HW2-4
Tutorial 2 Homework
Name
Graphing and data analysis
Tutorial section
AJITA: But look...because the ball is very bouncy, it hardly loses any velocity during the bounce.
That’s why it rebounds almost to its initial height. For instance, if it reached the floor
going down at 8.0 meters per second, then maybe it bounces back up at 7.9 meters per
second or even 7.95 meters per second. So, during the bounce, the velocity changes by a
tiny amount. Even though this change happens over a small time, the change in velocity
per change in time isn’t too big, probably smaller than when it’s just rising or falling.
Which student do you agree with? Explain.
Smitha is right. Ajita is talking about the change in speed, not the change in velocity. At the
bounce, the velocity suddenly jumps from a positive value, such as +8.0 m/s, to a negative value,
such as –7.9 m/s. The change in velocity, is therefore ∆v = vafter bounce – vbefore bounce = –7.9 m/s – 8.0
m/s = –15.9 m/s. That’s a large change in velocity to happen in such a short time. So, the
acceleration is large.
C. During the bounce, is the acceleration positive or negative? Explain.
Negative. As the above velocity vs. time graph shows, during the bounce, the velocity “jumps” down,
switching from positive to negative. So, that part of the velocity graph has a (huge) negative slope.
The same conclusion pops out of the calculation we did in part C. Since ∆v is negative, so is the
average acceleration, <a> = ∆v/∆t. Make sure you can hook up the graph and math.
© University of Maryland Physics Education Research Group, Fall 2004.
HW2-5