**Goal:** Reasoning with energy

**Source:** UMPERG-ctqpe64

Two

masses, M > m, travel down the surfaces shown. Both surfaces are

frictionless. Which mass has the largest speed at the bottom?

- m
- M
- Both have the same speed
- Cannot be determined

**Goal:** Reasoning with energy

**Source:** UMPERG-ctqpe64

Two

masses, M > m, travel down the surfaces shown. Both surfaces are

frictionless. Which mass has the largest speed at the bottom?

- m
- M
- Both have the same speed
- Cannot be determined

**Goal:** Reasoning with kinematics

**Source:** UMPERG-ctqpe63

Two

masses, M > m, travel down the surfaces shown. Both surfaces are

frictionless. Which mass has the largest __average__ speed during

their motion?

- m
- M
- Both have the same average speed
- Cannot be determined

(1) This problem is intended to promote discussion of average

speed. Both masses have the same speed at the bottom. Mass m has a

larger acceleration in the beginning because the circular track is

vertical at the outset. Although the angle of the incline is not

specified, the angle is irrelevant. All inclines will have the same

average speed. A simple graph of the speed of each mass versus time

shows that m will have the larger average speed.

**Goal:** Reasoning with energy

**Source:** UMPERG-ctqpe62

Two

identical blocks fall a distance H. One falls directly down, the other

slides down a frictionless incline. Which has the largest speed at the

bottom?

- The one falling vertically
- The one on the incline
- Both have the same speed
- Cannot be determined

(3) The only force doing work is gravity and both block undergo

the same vertical displacement.

**Goal:** Distinguish average velocity from velocity.

**Source:** UMPERG

A car is initially located at the 109 mile marker on a long straight

highway. Two and one half minutes later the car is located at the 111

mile marker.

What is the velocity of the car?

- 24 mph
- 32 mph
- 40 mph
- 48 mph
- 55 mph
- 64 mph
- Cannot be determined

The correct answer is (7) because only the average velocity can be

determined. However, students who respond (4) should not be

disconfirmed but prodded to be more discriminating when interpreting

questions. They have assumed that the car is traveling with a uniform

speed.

Students should be able to extract kinematical quantities from everyday

situations. They should also have a sense of the size of these

quantities.

What is the speed of the car when it is at the 109 mile marker? How do

you know?

Is it possible for the car to be at rest initially and reach the 111

mile marker two and one half minutes later? If it had constant

acceleration, what would its speed be when it reached the 111 mile

marker?

Have students make a sketch of position vs. time. They probably assume

that the speed is uniform throughout the time interval. Have them

consider other paths that still connect the two known points on the

position vs. time plot. Draw some reasonable path and have the students

describe what the car is doing during that interval.

**Goal:** Hone the concepts of speed and velocity.

**Source:** UMPERG

The radius of the Earth is 6,400 km. The speed and direction would you

have to travel along the equator to make the sun appear fixed in the sky

is most nearly

- 1680 km/hr, East
- 840 km/hr, East
- 533 km/hr, East
- 267 km/hr, East
- 267 km/hr, West
- 533 km/hr, West
- 840 km/hr, West
- 1680 km/hr, West
- Cannot be determined

(8) You would attempt to remain underneath the sun as it traveled from

East to West. Some students may be confused by the tilt of the Earth’s

axis and think that the Sun could not remain fixed in the sky if you

were constrained to move along the equator. These students would likely

answer (9).

Students should be able to determine the speed and direction even if

they do not yet have a solid grasp of velocity as a vector.

What is the circumference of the Earth? Does everyone on the Earth

travel at the same speed?

Build a simple model. Most students can readily grasp the result when

the Earth’s axis is perpendicular to the plane of the Earth’s orbit. A

model helps them understand that the tilt of the axis doesn’t matter.

**Goal:** Hone the vector nature of velocity.

**Source:** UMPERG-ctqpe142

A

child is standing at the rim of a rotating disk holding a rock. The

disk rotates without friction. The rock is thrown in the RADIAL

direction at the instant shown, which of the indicated paths most nearly

represents the path of the rock as seen from above the disk?

- path (1)
- path (2)
- path (3)
- path (4)
- path (5)
- cannot be determined

(4) is the correct path if the rock is thrown radially.

Once thrown the components of the velocity of the rock lying in a

horizontal plane are constant so the rock will have a path which is a

straight line.

Identify a coordinate frame. What are the components of the velocity

vector immediately after the rock is thrown?

What is the radial component of the velocity if the rock follows path

(2)?

Is it possible to throw the rock in such a way that the rock follows

path (5)?

This item should be compared to 63.

**Goal:** Understanding the first law.

**Source:** UMPERG-ctqpe120

A

child is standing at the rim of a rotating disk holding a rock. The

disk rotates without friction. If the rock is dropped at the instant

shown, which of the indicated paths most nearly represents the path of

the rock as seen from above the disk?

- path (1)
- path (2)
- path (3)
- path (4)
- path (5)
- cannot be determined

(2) is the correct path if the rock is simply dropped. Some students

selecting answer (3) may be viewing the rock from the child’s

perspective. Some students indicating choice (5) may interpret this

path as ‘straight down’.

This question is similar to others which seek to reveal student

perceptions about path persistence. It is a slightly different context

from the purely horizontal case of a ball rolling on a horizontal

surface around an semicircular section of hoop.

What path would the child see?

What is the velocity of the rock just before it is dropped? just after?

What would the path of the rock have been if the child continued to hold

it?

There are a variety of demonstrations that can be done as followup to

this question. It is important that students perceive the similarity

between the demonstration context and the problem situation.

**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?

- Object A only
- Object B only
- Object C only
- Both B and C
- Both A and C
- Both A and B
- All three have the same acceleration at t = 2.5s
- None of the above
- Cannot be determined

(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:** Associate velocity graph with physical motion.

**Source:** UMPERG

A soccer ball rolls across the road and down a hill as shown below.

Which of the following sketches of v_{x} vs. t represents the

horizontal velocity of the ball as a function of time?

(5) None of the above. The ball crosses the road in a straight line at

a more-or-less constant speed (perhaps slowing down slightly) provided

that the road is in good condition and the rolling friction between the

ball and road is sufficiently small. As the ball rolls down the hill it

will speed up, and so there will be an acceleration in the direction of

motion, with a non zero component to the right. The following graph is

a reasonable representation of the horizontal velocity as a function of

time.

This problem could challenge students in several areas: (1) Can

students recognize how the velocity is changing? What criteria do they

use? (2) Do students realize that as the ball moves down the hill it

speeds up and the x-component of velocity increases? Students may

associate the increase in velocity with the y-direction only. (3) Do

students associate the graph with the terrain over which an object

travels? The process of translation of a motion quantity to a graph can

be very difficult for students. (4) Will students confuse motion

quantities? When students analyze the graphs of velocity vs. time they

may be interpreting the graphs in terms of position instead of velocity.

Is the velocity ever zero? Where does the ball speed up? …slow down?

What is the direction of the velocity while the ball is on the sloped

section? Does the velocity have a non-zero horizontal component?

Set up a demonstration with a horizontal surface and a ramp, both with

the same net horizontal displacement. Roll a ball slowly across the

horizontal surface and down the ramp. Ask students to judge which

horizontal displacement took more time. Over what section (horizontal

surface or ramp) is the velocity larger on average?

**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.

## Commentary:

## Answer

(3) By energy considerations, both would have the same speed.

Students frequently get confused about the mass, thinking that the

larger mass has the greatest potential energy change and therefore has

the greatest speed.