Goal: Reasoning with dynamics.
Source: UMPERG
Two masses (M > m) are on an incline. Both surfaces have the same
coefficient of kinetic friction. Both objects start from rest, at the
same height. Which mass has the largest speed at the bottom?
Goal: linking acceleration and velocity graphically.
Source: UMPERG
The
plot of velocity versus time is shown at right for three objects. Which
object has the largest acceleration at t = 2.5s?
(6) Objects (A) and (B) have the same acceleration (i.e., they have the
same slope for the velocity vs. time graph at t=2.5s) Object (C) has a
constant velocity (zero slope).
After students have been introduced to acceleration, but before they are
given a procedure for determining the acceleration from a graph of
velocity vs. time. Students should answer this question after they have
gained an understanding of the definition of acceleration, but before
they are given any explicit instruction for how acceleration relates to
a velocity vs. time graph.
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 velocity vs. time graph indicate that an object
has a zero velocity? Zero acceleration? What features indicate a
negative acceleration? Positive acceleration?
Redraw the velocity vs. time graph for object (A) twice more. In one
drawing approximate the curve with three straight line segments. In the
second approximate the curve with 6 straight line segments.
Goal: Perceiving acceleration from description of motion.
Source: UMPERG Core A.4
How many of the objects below are NOT accelerating?
(A) A race car going around a circular track at 150 MPH
(B) A sky diver falling at a constant speed
(C) A heavy box sliding across the floor, after being released
(D) A bowling ball colliding with a pin
(E) A vibrating guitar string
(F) A baseball flying through the air
(G) A child swinging on a swing
(1) Only the skydiver, who has reached terminal velocity, has zero
acceleration. In each of the other situations either the speed or
direction of motion is changing.
Students may use a variety of factors in deciding whether an object
accelerates.
The goal is to focus students more narrowly on changes in the speed
and/or direction of an object’s motion.
How do you know whether an object is accelerating? What are some
examples of objects undergoing an acceleration?
Play a “challenge game” with the class where two teams of students take
turns challenging each other with situations in which an object
undergoes some motion, and the other team needs to determine whether or
not the object is accelerating.
Have students write out how they determine whether an object is
accelerating. After listening to the different method students use,
have students vote on which method they think is best.
002
Goal: Relating physical motion with graphical representation
Source: UMPERG
Which of the velocity vs. time plots shown below might represent the
velocity of a cart projected up an incline?
Select one of the above or:
(7) None of the above
(8) Cannot be determined
(3) or (4). Initially the cart has a non-zero velocity pointing up the
incline. The speed of the cart decreases as it moves up the incline,
reaching zero at its maximum height. The speed of the cart increases as
the cart moves down the incline. The velocity at the bottom of the
incline points down the incline. Graph (3)/(4) is correct if up/down
the incline is taken as the positive direction.
Students will often associate velocity time graphs with features of the
terrain. Many will pick either (5) because they neglect the vector
nature of velocity and think about the speed.
Is the velocity ever zero? Is the velocity ever positive? … negative?
When? Is the velocity constant? How do you know?
Plotting the position vs. time may help students come up with the
correct plot of velocity vs. time.
Goal: Honing the concept of acceleration especially regarding circular motion.
Source: UMPERG
A child is swinging. What is the direction of her acceleration when the
swing is at its lowest point?
(1) The acceleration is in the upward direction. The child is traveling
in a circle and at the lowest point the acceleration is all radial.
Circular motion must have been covered for the item to be of use. The
question may be answered using either kinematics or dynamics. The
direction of the acceleration can be realized using kinematics by
drawing the velocity vector just before the lowest point and just after
the lowest point. The difference is proportional to the acceleration
and this difference points toward the center of the circle. At the
lowest point all forces are vertical so the acceleration must also be in
the vertical direction. The tension is larger than the weight so the
acceleration is in the upward direction.
What is the definition of acceleration? What is the direction of the
velocity of the child when at the lowest point of the swing? Is it
getting larger or smaller? Is it changing direction? What forces act
on the child at the lowest point of the swing and in what direction are
these forces?
Have students draw a motion plot indicating the position and the
velocity vector of the child at various points in the child’s motion.
Do their drawings reflect that the velocity is always tangent to the
circular path, but increasing in magnitude as the child swings toward
the lowest point?
Goal: Relating physical understanding of an object’s behavior to a graphical representation of acceleration.
Source: UMPERG
A soccer ball rolls across the road and down a hill as shown below. At
the bottom of the hill the ball is given a quick kick so that the ball
goes back up the hill and across the road. The initial and final speed
of the ball is the same.
Which of the following sketches of ax vs. t is a reasonable
representation of the horizontal acceleration of the ball as a function
of time for period of time shown?
(2) The acceleration of the ball while on the slope is the same whether
it is going down or going up. Also, taking the positive direction to
the right, the kick would appear as a negative spike in the
acceleration.
This item is related to item 1. See comments there. Students need to
note that the plots are for the xcomponent of the acceleration.
How is the acceleration related to the velocity? Suppose the hill were
more inclined. What feature of the acceleration vs. time graph would
change? What is the direction of the velocity just before the kick?
just after?
Have students make a graph of velocity vs. time for each of the given
plots of acceleration vs. time. Have students generate a plot of the
acceleration and velocity in the y direction.
Goal: Differentiate between magnitude and direction of acceleration.
Source: UMPERG
| Case | Column 1 | Column 2 |
| (A) | A car goes from 0 to 60 mph in 6s along a straight highway. |
A car goes from 60 to 0 mph in 6s along a straight highway. |
| (B) | A race car travels around a circular track at 50 mph. |
A race car travels around the same circular track at 100 mph. |
| (C) | A ball is thrown straight up. It rises 20 ft. Ignore the effects of the air. |
A ball is dropped straight down. It falls 20 ft. Ignore the effects of the air. |
For which cases is the acceleration the same for the motion described
in both columns?
(3) The only case having the same acceleration is C where the
acceleration is that due to the gravitational force. In case A, the
magnitude of the two accelerations is the same but one is positive and
the other negative, i.e. the vectors point in opposite directions.
[This assumes that the acceleration is uniform.] In case B, the
“direction” is the same, i.e. pointing toward the center of the circle,
but the magnitudes are different.
This question reveals whether students have the concept of acceleration
as a vector (i.e. has direction as well as magnitude). Some students
may ignore the magnitude completely and key on the direction. The
objective here is to have students indicate the concept of acceleration
that they are using to answer the question.
For which cases is the magnitude of the acceleration the same? the
direction?
For which cases does the acceleration change during the motion
described?
Have students draw a motion diagram (a strobe diagram with the velocity
vector indicated at each position). This diagram helps students to
associate the acceleration with a change in velocity.
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?
(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.
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.
How is the average acceleration determined? What is the difference
between average acceleration and instantaneous acceleration? Where is
the instantaneous acceleration greatest?
Draw velocity vs. time graphs for the objects (A) and (B). Analyze the
average acceleration (instantaneous acceleration) for different time
intervals (times).
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 graph does not describe the situation. The acceleration is constant. The velocity is not zero at t=0s, but is zero at t=.9s.
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.
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?
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?
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?
(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.
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.
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.
Draw position vs. time graphs and velocity vs. time graphs for the
motion of objects that are difficult for students to analyze.
Commentary:
Answer
(3) Both will have the same speed. All the forces acting on the mass
(normal, friction, gravity) are proportional to the mass so the mass
cannot affect the acceleration experienced by the mass.