Author Archives: Automated Transfer Script

A2L Item 016

Goal: Differentiate between instantaneous and average acceleration.

Source: UMPERG

Below is shown a strobe diagram indicating the position of four objects
at successive time intervals. The objects move from left to right.

During the intervals shown, which object would you estimate has the
largest average acceleration?

  1. Object A
  2. Object B
  3. Object C
  4. Object D
  5. Objects A, B, & D
  6. Cannot estimate for (A) because its acceleration is changing
  7. Cannot estimate average acceleration from a strobe diagram
  8. None of the above
  9. Cannot be determined

Commentary:

Answer

(3) Assuming that the question is referring to magnitude, the largest
average acceleration is experienced by object (C). The other three
objects appear to start and end with approximately the same velocity.
For object (C) the velocity decreases in magnitude as the object moves
to the right. Students who answer (5) because they realize that the
average acceleration of C is negative and think zero is larger should
not be considered wrong.

Background

It is important for students to develop multiple ways of interpreting
concepts. This ensures that students are not just following rote
procedures to answer questions. Once an idea is understood students
should be able to use their understanding in a variety of contexts and
with a variety of representations.

The concept of average acceleration depends only on the initial and
final velocity over some specified time interval. Some students will
make their judgments on the basis of changes in the velocity at
different points in the motion.

Questions to Reveal Student Reasoning

How is the average acceleration determined? What is the difference
between average acceleration and instantaneous acceleration? Where is
the instantaneous acceleration greatest?

Suggestions

Draw velocity vs. time graphs for the objects (A) and (B). Analyze the
average acceleration (instantaneous acceleration) for different time
intervals (times).

A2L Item 014

Goal: Analyze and evaluate a solution to a given problem.

Source: UMPERG

A skateboarder heads straight up a steep bank angled at 45°, the whole time experiencing a constant acceleration. She manages to move 1.6m up the incline before rolling back down. The entire maneuver takes her 1.8 s, half of which is going up, the other half going down. What magnitude acceleration did she experience while on the incline?

Consider the steps in the following procedure. If the procedure is incorrect, respond with the number of the first incorrect step; if not, respond with step 7.

  1. The velocity vs. time graph for the situation is as shown.
  2. It takes the skateboarder 0.9 s to reach the highest point.
  3. The shaded area of the graph equals her displacement along the incline which is 1.6m.
  4. Equate this area (1/2 (0.9)v) to 1.6 and solve for v.
  5. Use v to find the slope of the velocity vs. time graph.
  6. The slope is equal to the acceleration.
  7. The procedure is correct.

Commentary:

Answer

(1) The graph does not describe the situation. The acceleration is constant. The velocity is not zero at t=0s, but is zero at t=.9s.

Background

This question requires students to make decisions and judgements which are needed when solving kinematics problems with understanding. This provides another opportunity to check students skills interpreting graphs and connecting the graph to the physical situation. Students may still be looking at superficial features of the graph to determine its validity.

Questions to Reveal Student Reasoning

Where is the skateboarder’s velocity zero? … velocity largest? Does this information match the graph?

What is her initial position? … her final position? Does this information match the graph?

Suggestions

Write out the appropriate solution plan. Ask students to compare the answers for the two approaches. Does an invalid plan necessarily lead to an incorrect answer? Why or why not? Does a valid plan necessarily lead to a correct answer? Why or why not?

A2L Item 013

Goal: Perceiving acceleration from changes in position.

Source: UMPERG

Below is shown a strobe diagram indicating the position of four objects
at successive (equal) time intervals. The objects move from left to
right.

During the intervals shown, which of the objects are accelerating?

  1. Object A only
  2. Object B only
  3. Object C only
  4. Object D only
  5. Objects A and B
  6. Objects B and C
  7. Objects A and C
  8. None of the above
  9. Cannot be determined

Commentary:

Answer

(7) (A) and (C) are clearly accelerating since the displacement is
different for different time intervals (implying different average
velocities). For (B) and (D) the average velocity is the same for each
time interval.

If there is something quirky about the motions of (B) and (D), it is
possible that these objects are accelerating even though their average
velocity is always the same for the time intervals observed. Therefore
students could be justified in selecting (9). Students should realize
that (A) and (C) are accelerating.

Background

It is important for students to develop multiple ways of interpreting
concepts. This ensures that students are not just following rote
procedures to answer questions. Once an idea is understood students
should be able to use their understanding in a variety of contexts and
with a variety of representations.

Questions to Reveal Student Thinking

Which objects have a non-zero velocity? How do you know? How can you
determine from an object’s position at several times whether it is
accelerating? What features of a strobe diagram indicate that an object
has a non-zero velocity? a non-zero acceleration?

What are some physical situations that correspond to the different
motions in the strobe diagram.

Suggestions

Draw position vs. time graphs and velocity vs. time graphs for the
motion of objects that are difficult for students to analyze.

A2L Item 011

Goal: Analyze and evaluate a solution to a given problem.

Source: UMPERG

In order to solve the problem:

An airplane accelerates down a runway in order to take off
but aborts and applies brakes causing the plane to stop. The plane
speeds up at a constant rate for 5 seconds, then slows down at the same
rate when the brakes are applied. The plane stops at a point that is
100 meters from its initial position. What was the acceleration of the
airplane during the first 5 seconds?

Someone suggests the following procedure:

(A) The acceleration of the plane is constant and the same for the
entire motion.

(B) The entire process takes 10 seconds and the displacement is 100
meters.

(C) It is possible, therefore, to use the formula “change in x =
vo,x t + 1/2 ax t2“, where
vo,x is zero and t = 10s.

(D) The only unknown in this equation is ax, so solve for it.

Which of the following is true?

  1. The procedure is invalid because statement A is incorrect.
  2. The procedure is invalid because statement B is incorrect.
  3. The procedure is invalid because statement C is incorrect.
  4. The procedure is invalid because statement D is incorrect.
  5. The procedure is invalid because more than one statement is incorrect.
  6. The procedure is valid.

Commentary:

Answer

(5) More than one statement is incorrect. The
acceleration is not constant for the entire motion and so (A) is
incorrect. Although the magnitude of the acceleration is constant its
direction changes. Statement (C) is incorrect because the formula in
(C) is only valid over periods the acceleration is constant.

Background

This question requires students to make decisions and judgements which
are needed when solving kinematics problems with understanding. The
kinematics equations are of limited use. They apply directly only to
problems involving constant acceleration. Students are usually not
aware of this limitation and are apt to apply the kinematics expressions
much too broadly. Students also tend to view acceleration as a scalar
quantity and therefore see the acceleration as constant even when it is
not so.

Questions to Reveal Student Thinking

How do we determine the acceleration. What is the acceleration while
the plane is speeding up? … slowing down? If necessary ask the
following. What is the direction of acceleration while the plane is
speeding up? … slowing down?

Suggestions

Draw a graph of velocity vs. time for constant acceleration. Draw a
graph of velocity vs. time for the problem situation. Discuss the
acceleration and displacement in terms of these graphs.

A2L Item 012

Goal: Recognizing acceleration in graphs of position vs. time

Source: UMPERG

The plot of position versus time is shown for three objects. Which object has the largest acceleration at t = 2.5s?

  1. Object A only
  2. Object B only
  3. Object C only
  4. Both B and C
  5. Both A and C
  6. Both A and B
  7. All three have the same acceleration at t = 2.5s
  8. None of the above
  9. Cannot be determined

Commentary:

Answer

(1); the slope of position v. time is velocity, and the slope of
velocity v. time is acceleration. The only plot with a non-zero
(positive) acceleration is A. Plot C denotes zero velocity, and plot B
denotes a constant velocity.

Background

It is important for students to develop multiple ways of interpreting
concepts. Graphical representations are often more useful than
algebraic representations in solving kinematic problems. In this
instance students must recognize the signature of acceleration in a plot
of position vs time.

Questions to Reveal Student Thinking

How can you determine if an object is accelerating? For which objects
is the velocity changing. What are some examples of objects moving
according to the graphs?

What features about a position vs. time graph indicate that an object
has a zero velocity? a zero acceleration? What features indicate a
negative acceleration? a positive acceleration?

Suggestions

Draw a graph of velocity vs. time for each object. Then draw the graph
of acceleration vs. time.

Follow up question: Can the position vs. time of an object have a
negative slope at some specific time, and yet the acceleration be
positive at that same time?

A2L Item 010

Goal: Recognize sources of measurement error.

Source: UMPERG

Using a stopwatch to measure the time for a ball to fall a distance of 1
meter, and based upon 3 measurements of the time, a student obtained a
value for the gravitational constant of g = 9.4 +/- 0.2 m/s2.
What can be done to improve the accuracy of the data?

  1. Take more data
  2. Allow the ball to fall a larger distance
  3. Use a digital timer
  4. Allow the ball to roll down a slope rather than freefall
  5. Use a heavier ball
  6. More than one, but not all, of the above would improve the accuracy
  7. All of the above would improve the accuracy
  8. None of the above would improve the accuracy
  9. Cannot be determined

Commentary:

Answer

Several factors, such as air resistance and reaction time, can affect
the accuracy of the measurement. Without more details, we do not know
which factors have the largest effect. For example, if the ball falls
over a larger distance the factor of reaction time becomes less
significant, but the factor of air resistance becomes more significant.
The impact on the overall accuracy is undetermined. A ball rolling
down a slope would move slower and would decrease the effects of air
resistance and reaction time. But new factors such as rolling friction
and moment of inertia (radius) need to be considered. Students will
make assumptions. Getting students to state their assumptions is what
is most important.

Background

Issues to consider: (1) Do students think that merely collecting more
data or using a digital timer will improve accuracy? (2) Can students
identify factors that could impact the accuracy of measuring the
gravitational constant using the method described? (3) Can students
describe how the factors impact the accuracy and how the proposed
modifications would increase/decrease the accuracy.

Questions to Reveal Student Thinking

How do we know that the measured value for g is inaccurate? What factors
impact the accuracy? What factor impacts the accuracy the most?

Does increasing the distance the ball falls impact the accuracy? Why?
How?

Consider some of the other factors.

Suggestions

Have different groups of students measure the time for an object to fall
one meter. Have each group consider a different object: (1) crumpled
paper, (2) small metal ball, (3) two objects of the same size and shape,
but different weight, etc.

A2L Item 009

Goal: Relate friction, velocity, and time.

Source: UMPERG

A cart rolls across a table two meters in length. Half of the length of
the table is covered with felt which slows the cart at a constant rate.
Where should the felt be placed so that the cart crosses the table in
the least amount of time?

  1. On the first half of the table
  2. Centered on the table
  3. On the second half of the table
  4. It doesn’t matter where the felt is placed
  5. None of the above
  6. Cannot be determined

Commentary:

Answer

The
felt should be placed on the second half of the table. After the cart
rolls across the felt it will travel at a lower speed. To minimize the
time to cross the table one must minimize the time the cart spends at
the lower speed. The graph to the right illustrates the point for the
two extreme cases: felt on first half (gray curve) and felt on second
half (black curve). The velocity vs. time graph for the case where the
felt is on the second half of the table is above the velocity vs. time
graph for all other possibilities. Answer (3) is the best choice.

Background

Students should have some experience using the concepts of velocity and
acceleration to solve kinematics problems and analyze graphs. The
question students need to answer is what configuration will permit the
cart to travel at a higher speed for the longest period of time (or the
lowest speed for the shortest period of time). A graph provides support
for a conceptual argument.

Issues to consider: (1) Can students reason and analyze a situation
involving constant acceleration. (2) Do students try to solve the
problem using algebraic methods? (2) Can students use graphical methods
and conceptual reasoning? (3) Can students verbalize the central idea —
an object will travel a certain distance in less time if its speed is
higher?

Questions to Reveal Student Thinking

Where is the cart moving the fastest? … the slowest?

What does a graph of the velocity vs. time look like?

How do you determine when the cart has reached the end of the table from
a graph of velocity vs. time?

Suggestions

Try some limiting cases. If the piece of felt were small (say 10 cm)
but slowed the cart from 1 to .8 m/s on a 3m table. Approximately how
long would the trip take if the felt were placed at the beginning of the
table?…at the end of the table?

A2L Item 008

Goal: Relate acceleration to slope of velocity graph

Source: UMPERG

A toy rocket blasts off with an acceleration of 9.8 m/s2
upward. After 5 seconds the rocket releases its “payload” but continues
to accelerate upward at the same rate. In the following graph the solid
line represents the velocity of the rocket as a function of time and the
dashed line represents the velocity of the payload.

Which of the following statements regarding this situation are correct?

A. The payload hits the ground at t = 10s.

B. The slope of the rocket’s velocity vs. time graph is 9.8 m/s2.

C. The velocities of the rocket and the payload point in opposite
directions after the payload is released.

D. The area of the shaded region can be determined from the given
information.

E. The payload spends half the time in the air as the rocket.

F. The payload is released at 1/2 the maximum height of the rocket.

  1. Only (A)
  2. Only (B)
  3. Only (C)
  4. Only (D)
  5. Only (E)
  6. Only (F)
  7. Two of the statements are true
  8. Three of the statements are true
  9. None is true

Commentary:

Answer

Answer (7) is the best choice. The only statements that are true are
(B) and (D). The height of the rocket in relation to the height of the
payload can be determined by the ratio of the areas under their
respective velocity vs. time graphs. The payload reaches its maximum
height at t=10s (i.e., when its velocity is equal to zero). At t=10s the
area under the rocket’s v vs. t graph is twice the area under payload’s
v vs. t graph. This can be determined without knowing any of the
information contained in (A) through (F).

Background

Issues to consider: (1) Can students identify and evaluate information
needed to judge the correctness of a statement? (2) Can students
interpret a graphical representation of information? (3) Can students
determine the point of maximum displacement from a v vs. t graph? (4)
Can students interpret the significance of slope and area for a v vs. t
graph. (5) Do students confuse v vs. t graphs with x vs. t graphs.

Questions to Reveal Student Thinking

When does the payload start to fall back toward the earth? When does
the payload hit the earth? What is the acceleration of the payload
after it is released?

Which of the statements are true? Explain. Which of the statements are
false? Explain.

Could the true statements be used to determine the heights of the rocket
and payload? Explain.

Suggestions

As a class draw a rough strobe diagram. Relate times in the strobe
diagram to times on the graph.

Discuss with students how they would approach the problem algebraically.

A2L Item 007

Goal: Linking acceleration to changes in velocity.

Source: UMPERG

A marble rolls onto a piece of felt that is 30 cm in length. At 20 cm
the speed of the marble is half of its initial value. Which of the
following is true? Assume that the acceleration is constant on the felt.

  1. The marble will come to rest on the felt.
  2. The marble will go past the end of the felt.
  3. What will happen cannot be determined.

Commentary:

Answer

(1) The marble will come to rest on the felt. A graph of velocity vs. time is helpful for analyzing this problem. The distance traveled while slowing down to half its initial speed (i.e., the first 20 cm) is three times the additional distance (i.e., the distance beyond 20 cm) the marble will roll before coming to rest. This can be seen by comparing the areas for
these two different time periods. The marble will come to rest at approximately 26.7 cm.

Background

Students should have some experience using the concept of acceleration to solve kinematics problems and analyze graphs. The answer is less important than how students represent the problem and how they approach solving the problem.

Issues to consider:
(1) Do students only solve the problem using algebraic methods? (2) To what extent do students use other approaches? (3) Do students use graphical methods involving areas? (4) Do theycompare average speeds for the two periods (i.e., the period covering the first 20 cm and the remaining period of time before the marble comes to rest)? (5) Do they compare the actual speeds of the marble at each instant of time for the two time periods (the ratio is usually greater than or equal to 2/1 at each corresponding time, as shown in the accompanying graph). (6) Even if the students use algebraic methods, do they employ a strategy or do they do so mindlessly?

Questions to Reveal Student Thinking

Ask students to consider the following context (which they are familiar with and is algebraically simple): an object is dropped from rest. How fast is it moving after one second? … after two seconds? …after three seconds? How far has it traveled after one second? …after two seconds?…after three seconds? What is the relationship between velocity and position? Why is the relationship not linear?

Suggestions

If students do not use a graph to solve the problem, ask them to draw a velocity vs. time graph for the situation and then use the graph to solve the problem.

A demonstration is possible.

A2L Item 006

Goal: Contrast instantaneous and average acceleration. Explore the difference between them.

Source: UMPERG

Which of the following motions has a zero value for the average
acceleration during the specified time interval?

A. A race car makes one lap around an oval track at constant speed.
The time interval is the time to complete the lap.

B. A cart collides with a wall and rebounds with the same speed. The
time interval is the time during which the cart is in contact with the
wall.

C. A ball rolls up, then down a hill. The time interval is the time
the ball is on the hill.

  1. Only (A)
  2. Only (B)
  3. Only (C)
  4. Both (A) and (B)
  5. Both (A) and (C)
  6. Both (B) and (C)
  7. All three motions have zero average acceleration
  8. None has zero average acceleration
  9. Can’t be determined

Commentary:

Answer

Situation A is the only one where the velocity is the
same at the beginning and end of the time interval. Therefore, it is
the only situation where the change in velocity is zero. Answer (1) is
the best choice.

Background

Use this item early in the
study of acceleration. It requires that students be familiar with the
definition of average acceleration. Issues: (1) Do students have a
working knowledge of the vector nature of velocity and acceleration?
(2) Do students incorrectly think that the average acceleration must be
zero if an object’s speed is the same at the beginning and end of the
time interval, even if the direction of the velocity vector at these two
times is different. (3) Do students correctly perceive that the
velocity is different when an object changes direction?

Questions to Reveal Student Reasoning

  • What is the definition of average
    acceleration? How do you compute the average acceleration for a given
    time interval?

  • How can you tell if the velocity has changed? If an object has the same
    speed at two different times, is the average acceleration necessarily
    zero? Why, or why not?

  • What are some good rules of thumb for ascertaining whether or not the
    average acceleration of an object is zero for a given time interval?

Suggestions

Have students write out examples of motion
for a finite time interval. Have students present their examples to the
class. Have the class decide whether the average acceleration is zero.

Let the class devise ways of determining whether the average
acceleration in a certain situation is zero.