AP Physics 1 Unit 3: Work, Energy, and Power

Use work and energy models to connect kinetic and potential energy, system boundaries, conservation laws, nonconservative interactions, and power.

Covered syllabus points

3.1 Translational Kinetic Energy 3.2 Work 3.3 Potential Energy 3.4 Conservation of Energy 3.5 Power

[Maximum number: 3]

(12 points, suggested time 25 minutes)
A student hangs a spring of unknown spring constant k vertically by attaching one end to a stand, as shown in Figure 1. The other end of the spring has a small loop from which small cylinders can be hung. In addition to the spring, the student has access only to a variety of cylinders of unknown masses, a stopwatch, and a digital scale.

Using the data in the velocity-time graph, calculate the change in kinetic energy of the cart from $t=0.5 \mathrm{~s}$ to $t=2.0 \mathrm{~s}$ . Show your steps and substitutions.

(i)
For using the kinetic energy equation $K=\frac{1}{2} m v^{2}$ 1 point
For substituting one of the following: 1 point

- 0.25 kg as the mass

- $0.3 \mathrm{~m} / \mathrm{s}$ as the initial velocity

For an answer that approximates the change in kinetic energy to be $\Delta K \approx-0.0113 \mathrm{~J}$ 1 point

Scoring Note: A correct response with no supporting work earns this point only.
Scoring Note: The unit and the negative sign are not required to earn this point.
Example Response

\begin{aligned} & \Delta K=K_{f}-K_{i} \\ & \Delta K=\frac{1}{2} m v_{f}^{2}-\frac{1}{2} m v_{i}^{2} \\ & \Delta K=\frac{1}{2} 0.25(0)^{2}-\frac{1}{2} 0.25(0.3)^{2} \\ & \Delta K=-0.0113 \mathrm{~J} \end{aligned}

[Maximum number: 12]

(12 points, suggested time 25 minutes)
A block is initially at position x=0 and in contact with an uncompressed spring of negligible mass. The block is pushed back along a frictionless surface from position x=0 to x=-D, as shown above, compressing the spring by an amount $\Delta x=D$ . The block is then released. At x=0 the block enters a rough part of the track and eventually comes to rest at position x=3 D. The coefficient of kinetic friction between the block and the rough track is $\mu$ .

Distribution

of points
(a) 4 points

For sketching either energy curve with a reasonably correct shape between
x
D
= -
and
0
x =, with zero and maximum values at the correct locations

1 point
For sketching two curves from x
D
= -
to
0
x = with shapes and values such that the total energy is constant (even if the curves are incorrect)

1 point
For sketching potential energy equal to zero from
0
x =
to
3
x
D
=

1 point
For sketching kinetic energy as a linear function from its maximum value at
0
x =

to zero at
3
x
D
=

1 point

(b)
(i) 1 point

For identifying that the student is correct that the block will have more energy when it leaves the spring

1 point

(ii) 1 point

For identifying that the student is incorrect about the new final position of the block because the spring’s energy does not scale linearly with its compression

1 point

AP® PHYSICS 1

Distribution

of points

(c) 3 points

For indicating that the final energy in the spring (which becomes the mechanical energy of the block as it reaches the rough track) is four times the original energy in the spring

1 point
For indicating that the frictional force remains the same

1 point
For equating the initial energy in the spring to an expression that shows that the energy dissipated by friction is proportional to the distance the block slides
down the rough track

1 point
Example:

2
1
1
2
U
kD
=
and
(
)2
2
1
2
2
U
k
D
=
so
2
1
4
U
U
=

(
)
1
3
W
mg
D
m
=
and
2
2
W
mg x
m
D
=

1
1
W
U
=
and
2
2
1
1
4
4
W
U
U
W
= = =

(
)
(
)
2
4
3
mg x
mg
D
m
m
D
=

(
)
2
4 3
12
x
D
D
D
= =

(d) 3 points

For indicating that the student’s correct reasoning that the block has more energy
in the second situation is expressed by the calculations comparing the initial energy in the spring

1 point
For indicating that the student’s correct reasoning that the block will slide farther
is expressed by an equation that indicates that the work done by friction to the block in the second situation is some factor greater than the work
done in the first situation

1 point
For indicating that the student’s incorrect reasoning that energy scales linearly with the spring’s compression is corrected by the expression for the initial energy of the spring

1 point

AP® PHYSICS 1

(a)

On the axes below, sketch and label graphs of the following two quantities as a function of the position of the block between x=-D and x=3 D. You do not need to calculate values for the vertical axis, but the same vertical scale should be used for both quantities.

The kinetic energy K of the block

The potential energy U of the block-spring system

Figure for Question (a) — AP Physics 1: Algebra-Based

The spring is now compressed twice as much, to $\Delta x=2 D$ . A student is asked to predict whether the final position of the block will be twice as far at x=6 D. The student reasons that since the spring will be compressed twice as much as before, the block will have more energy when it leaves the spring, so it will slide farther along the track before stopping at position x=6 D.

[ 4 ]

(b)

Which aspects of the student's reasoning, if any, are correct? Explain how you arrived at your answer.

[ 1 ]

(c)

Which aspects of the student's reasoning, if any, are incorrect? Explain how you arrived at your answer.

[ 1 ]

(d)

Use quantitative reasoning, including equations as needed, to develop an expression for the new final position of the block. Express your answer in terms of D.

[ 3 ]

3 points

For indicating that the final energy in the spring (which becomes the mechanical energy of the block as it reaches the rough track) is four times the original energy in the spring
For indicating that the frictional force remains the same
For equating the initial energy in the spring to an expression that shows that the energy dissipated by friction is proportional to the distance the block slides down the rough track
Example:

\begin{aligned} & U_{1}=\frac{1}{2} k D^{2} \text { and } U_{2}=\frac{1}{2} k(2 D)^{2} \text { so } U_{2}=4 U_{1} \\ & W_{1}=\mu m g(3 D) \text { and } W_{2}=\mu m g \Delta x_{2} \\ & W_{1}=U_{1} \text { and } W_{2}=U_{2}=4 U_{1}=4 W_{1} \\ & \mu m g \Delta x_{2}=4(\mu m g(3 D)) \\ & \Delta x_{2}=4(3 D)=12 D \end{aligned}

(e)

Explain how any correct aspects of the student's reasoning identified in part (b) are expressed by your mathematical relationships in part (c). Explain how your relationships in part (c) correct any incorrect aspects of the student's reasoning identified in part (b). Refer to the relationships you wrote in part (c), not just the final answer you obtained by manipulating those relationships.

Figure for Question (e) — AP Physics 1: Algebra-Based

[ 3 ]

[Maximum number: 12]

A block of mass M is released from rest at position x=0 near the top of a ramp. The ramp

makes an angle of $\theta$ with the horizontal. The block slides down the ramp with negligible

friction. At x=8 D the block makes contact with an uncompressed spring with spring

constant k. The spring is then compressed and the block momentarily comes to rest at x=12 D.

Figure 1 shows the instants when the block is at x=0, x=6 D, and x=10 D, respectively.

Figure for Question 3 Work, Energy, and Power question 3 — AP Physics 1: Algebra-Based

(a)

Figure 4 shows an energy bar chart that represents the kinetic energy K of the block, the

gravitational potential energy $U_{g}$ of the block-spring-Earth system, and the spring potential

energy $U_{s}$ of the block-spring-Earth system at the instant that the block is at x=10 D. The

gravitational potential energy $U_{g}$ of the block-spring-Earth system is defined to be zero when

the block momentarily comes to rest at x=12 D.

Draw shaded bars that represent $K, U_{g}$ , and $U_{s}$ to complete the energy bar charts in

Figure 2 and Figure 3 for when the block is released from rest at x=0 and for when the

block is at x=6 D, respectively.

- Shaded bars should start at the dashed line that represents zero energy.

- Represent any energy that is equal to zero with a distinct line on the zero-energy line.

- The relative heights of each shaded bar should reflect the magnitude of the respective

energy consistent with the scale used in Figure 4.

Figure 2

Figure 3

Figure 4

[ 3 ]

A	For drawing one bar in Figure 2 that shows only gravitational potential energy $\left(U_{g}\right)$ <br>Scoring Note: The correct height of the bar is not required to earn this point.	Point A1
	For including only K and $U_{g}$ in Figure 3<br>Scoring Note: The correct heights of the bars are not required to earn this point.	Point A2
	For drawing bars in figures 2 and 3 whose total height, respectively, equals $12 E_{0}$	Point A3
Example Response
	Energy x=6 D<br> $14 E_{0}$ <br> $12 E_{0}$ <br> $10 E_{0}$ <br> $8 E_{0}$ <br> $6 E_{0}$ <br> $4 E_{0}$ <br> $2 E_{0} \pm \cdots$ <br> ${ }_{0} \pm=\cdots$ <br>\hdashline $U_{g}$
	Figure 2

(b)

Figure 5 shows the block at x=0 when the block is released from rest and the block at

x=12 D when the block momentarily comes to rest against the compressed spring.

Figure 5

Starting with conservation of energy, derive an equation for the spring constant k. Express

your answer in terms of $M, \theta, D$ , and physical constants, as appropriate. Begin your

derivation by writing a fundamental physics principle or an equation from the reference

information.

[ 4 ]

B	For a multistep derivation that includes conservation of energy	Point B1
		For equating the gravitational potential energy to the spring potential energy	Point B2
		For one of the following: - Substituting $12 D \sin \theta$ for $\Delta y$ in a gravitational potential energy expression - Substituting 4 D in for $\Delta x$ in a spring potential energy expression	Point B3
		For a correct expression for k<br>Accept any algebraic equivalent of one of the following: - $k=\frac{3}{2} \frac{M g \sin \theta}{D}$ - $\frac{3}{2} \frac{M g \sin \theta}{D}$	Point B4
		Example Response $\begin{aligned} & E_{0}=E_{f} & U_{g}=U_{s} & m g \Delta y=\frac{1}{2} k(\Delta x)^{2} & M g(12 D \sin \theta)=\frac{1}{2} k(4 D)^{2} & k=\frac{3}{2} \frac{M g \sin \theta}{D} \end{aligned}$

(c)

Figure 6 shows a graph of the energy of the system as a function of the position of the block

from x=8 D to x=12 D. The spring potential energy $U_{s}$ of the block-spring-Earth system is

shown on the graph.

On the axes shown in Figure 6, do the following.

[ 3 ]

(i)

Sketch and label a line or curve that represents the total mechanical energy E for the

block-spring-Earth system as a function of the position of the block from x=8 D

to x=12 D.

[ 1 ]

C (i) For sketching a horizontal line at $12 E_{0}$ , continuous from 8 D to 12 D, with a label that

indicates total mechanical energy

(ii)

Sketch and label a line or curve that represents the gravitational potential energy $U_{g}$ for

the block-spring-Earth system as a function of the position of the block from x=8 D

to x=12 D.

Figure 6

[ 2 ]

(ii) For sketching a straight line that is decreasing with a label that indicates gravitational

Point C2

potential energy

For starting the line that represents gravitational potential energy at the point ( $8 D, 4 E_{0}$ )

Point C3

Example Response

(d)

Indicate whether the speed $v_{9 D}$ of the block at x=9 D is greater than, less than, or equal to

the speed $v_{8 D}$ of the block at x=8 D.

$v_{9 D}>v_{8 D}$

$v_{9 D}<v_{8 D}$

$v_{9 D}=v_{8 D}$

Justify how your response is consistent with the energy lines or curves you drew in Figure 6

in part C.

[ 2 ]

D For one of the following:

- Indicating $v_{9 D}>v_{8 D}$ if the graph in part C shows

$U_{g}+U_{s} \text { at } x=8 D \text { is greater than } U_{g}+U_{s} \text { at } x=9 D$

- Indicating $v_{9 D}<v_{8 D}$ if the graph in part C shows

$U_{g}+U_{s} \text { at } x=8 D \text { is less than } U_{g}+U_{s} \text { at } x=9 D$

- Indicating $v_{9 D}=v_{8 D}$ if the graph in part C shows

$U_{g}+U_{s} \text { at } x=8 D \text { is equal to } U_{g}+U_{s} \text { at } x=9 D$

For a justification consistent with the graph that correctly relates the speed to the

Point D2

gravitational potential energy, spring potential energy, and either kinetic energy or total

energy

Example Response

The speed at x=9 D is greater than the speed at x=8 D. From the graph, the kinetic

energy at x=8 D must be $8 E_{0}$ . At x=9 D, the spring potential energy is $<E_{0}$ and

the gravitational potential energy is $3 E_{0}$ , leaving $>8 E_{0}$ of the energy as kinetic.

Therefore, the speed must be greater at x=9 D than at x=8 D.

[Maximum number: 12]

Figure for Question 3 Work, Energy, and Power question 4 — AP Physics 1: Algebra-Based

(12 points, suggested time 25 minutes) Food scientists have created a new oil. At room temperature, the oil is a liquid. As the oil gets colder, however, it stiffens (thickens) into a sticky gel. To explore the properties of the oil, the scientists fill a container with the oil to a height D, as shown in the figure above on the left. They drop a small steel ball of mass M from rest at the top of the oil. Using video to capture the ball's motion, the scientists calculate $E_{\text {lost }}$ , the mechanical energy lost by the ball-Earth system from the time the ball enters the oil to the time just before the ball strikes the bottom of the container. The scientists also define the "stiffness" S of the oil as a quantity proportional to the force required to move a rod through the oil at a standard constant speed. The scientists calculate $E_{\text {lost }}$ and S at several different temperatures, ranging from room temperature to the lowest temperature at which the ball still falls through the oil. The graph above on the right shows $E_{\text {lost }}$ as a function of S for the calculated data points and a best-fit curve.

(a)

(a) Give a physical reason why the curve in the graph would not reach the vertical $\left(E_{\text {lost }}\right)$ axis even if the scientists had taken data over a broader range of temperatures.

[ 2 ]

(b)

Figure for Question (b) — AP Physics 1: Algebra-Based

(b) As S increases, $E_{\text {lost }}$ approaches a maximum value labeled $E_{\text {max }}$ on the graph above. Write an equation for $E_{\text {max }}$ in terms of M, D, and physical constants, as appropriate. Justify your answer.

[ 3 ]

(b) 3 points

For a correct equation: $E_{\text {max }}=M g D$ OR $E_{\text {max }}=-M g D$

For stating that the maximum energy that can be lost is equal to the initial energy of the

system

For stating that the initial energy of the system is the initial gravitational potential energy

Example: The maximum energy that can be lost is equal to the initial energy, which is all

gravitational potential energy: M g D.

Alternate solution for the last 2 points

For a statement or the application of mechanical energy as the sum of kinetic and

potential energies

For combining the change in the gravitational potential energy with a small change in

kinetic energy as the ball falls to make a valid argument for $E_{\max }$ .

Example: In a very stiff oil that the ball can barely fall through, it gains essentially no

kinetic energy. In this case the overall change in mechanical energy, $\Delta K+\Delta U$ , is

approximately equal to $\Delta U$ because $\Delta K$ is essentially zero.

So, $E_{\text {max }}=|\Delta U|=M g D$ .

(c)

Figure for Question (c) — AP Physics 1: Algebra-Based

(c) One of the scientists, in trying to represent the relationship between the oil stiffness and the mechanical energy lost, writes down the equation $E_{\text {lost }}=C S^{2}$ , where C is a constant with appropriate units. Another scientist points out that this equation cannot be correct. Give two reasons why the equation cannot be correct.

[ 2 ]

For each of the following, up to a maximum of 2 points

1 point for each valid, independent statement that the equation does not match the graph

(maximum of 2 points).

1 point for a statement that demonstrates that $E_{\text {lost }}$ cannot increase indefinitely.

Example statements ( 1 point each, any two could be combined for 2 points):

The equation indicates that $E_{\text {lost }}$ increases indefinitely with S, but the graph has an

upper limit for $E_{\text {lost }}$ .

The equation represents a parabola (or is concave-up), which does not match the

graph.

It doesn't make physical sense that $E_{\text {lost }}$ could be more than the available

potential energy MgD.

(d)

Figure for Question (d) — AP Physics 1: Algebra-Based

(d) Further attempting to model the ball's motion, the scientists write the following equation for the time t the ball takes to fall through the oil: $t=\frac{Z}{S}$ , where Z is a constant with appropriate units. Is this equation plausible-in other words, does it make physical sense? Plausible Not plausible Briefly explain your reasoning.

[ 2 ]

(e)

(e) $\Delta K$ is the change in kinetic energy of the ball between the time it is released from rest and the time just before the ball strikes the bottom of the container. On the axes below, sketch $\Delta K$ as a function of S, the oil stiffness.

[ 3 ]

(e) 3 points

Figure for Answer to Question (e) — AP Physics 1: Algebra-Based

For drawing a concave-upward curve that approaches the horizontal at higher S

For having the maximum value of $\Delta K$ at the lowest graphed value of S, which can be

S=0 or a positive value

Note: This point is not earned if the graph indicates that $\Delta K$ approaches infinity as S

approaches zero.

For high values of S, the graph approaches and does not cross the horizontal axis

Note: This point is earned if the graph touches the horizontal axis without crossing it.

All question bank results loaded

Covered syllabus points

3 Work, Energy, and Power question 1

3 Work, Energy, and Power question 2

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)

3 Work, Energy, and Power question 3

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

Question (d)

3 Work, Energy, and Power question 4

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)