AP Physics 1 Unit 4: Linear Momentum

Analyze interactions with momentum and impulse, choose appropriate systems, and distinguish elastic from inelastic collisions using energy and momentum.

[Maximum number: 1]

The magnitude of the one-dimensional momentum of a 2-kilogram particle obeys the relationship p=2 t+3. What was the velocity of the particle at t=1 second?

A

5 m/s

B

2 m/s

C

1 m/s

D

2.5 m/s

[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.

(a)

Using the data in the force-time graph, estimate the change in momentum of the cart from $t=0.5 \mathrm{~s}$ to $t=2.5 \mathrm{~s}$ . Briefly explain how you arrived at your estimation.

[ 2 ]

(b)

Do the data from the velocity-time graph confirm your estimation from part (c)(ii) ? Briefly explain.

Figure 1

[ 1 ]

(iii) For an explanation that compares the estimated value of the change in momentum from 1 point (c)(ii) to the data from the velocity-time graph

Example Response
The velocity-time graph shows that velocity is $0.3 \mathrm{~m} / \mathrm{s}$ at both 0.5 s and 2.5 s, and momentum is mass times velocity, so the momentum is the same at both times. This agrees with my estimation from part (c)(ii) that the change in momentum is zero. for question 212 points

[Maximum number: 10]

A student has a cart of mass $m_{\mathrm{c}}$ and a block of mass $\frac{1}{5} m_{\mathrm{c}}$ , as shown in Figure 1.

- At time t=0, the cart is moving to the right across a horizontal surface with constant

speed $v_{\mathrm{c}}$ , and the student releases the block from rest.

- At $t=t_{1}$ , the block collides with and sticks to the top of the cart. The block does not slide

on the cart.

- At $t=t_{2}$ , the block-cart system continues to move to the right with constant speed $v_{f}$ .

Figure 1

(a)

A.

[ 7 ]

(i)

On the axes shown in Figure 2, sketch a graph of the magnitude $p_{x}$ of the x-component

of the momentum of the block-cart system as a function of time t from t=0 until $t>t_{2}$ .

Figure 2

[ 2 ]

A (i) For sketching one of the following:

- A constant $p_{x}$ for $t<t_{1}$

- A constant $p_{x}$ for $t>t_{1}$

For sketching a line that demonstrates momentum is constant

Point A2

Accept one of the following:

- A continuous, nonzero, horizontal line from t=0 to $t=t_{2}$

- A continuous, nonzero, horizontal line from t=0 to $t>t_{2}$

Example Response

Official scoring-guideline example response

(ii)

Derive an expression for the speed $v_{f}$ of the block-cart system after time $t=t_{2}$ in terms

of $m_{\mathrm{c}}, v_{\mathrm{c}}$ , and physical constants, as appropriate. Begin your derivation by writing a

fundamental physics principle or an equation from the reference information.

[ 2 ]

(ii) For including a conservation of momentum equation

Point A3

Scoring Note: Part A (ii) and part A (iii) may be scored together, if necessary.

For setting $m_{\mathrm{c}} v_{\mathrm{c}}$ equal to the momentum of the block-cart system after the collision

Point A4

Scoring Note: A correct, isolated, final expression of $v_{f}=\frac{5}{6} v_{\mathrm{c}}$ earns points A3 and

A4.

Example Response

$\begin{aligned} & p_{i}=p_{f} \\ & m_{\mathrm{c}} v_{\mathrm{c}}=\left(m_{\mathrm{c}}+m_{\mathrm{b}}\right) v_{f}=\left(m_{\mathrm{c}}+\frac{1}{5} m_{\mathrm{c}}\right) v_{f} \\ & v_{f}=\frac{5}{6} v_{\mathrm{c}} \end{aligned}$

(iii)

Derive an expression for the change in the kinetic energy $\Delta K$ in the block-cart system

from t=0 to $t=t_{2}$ in terms of $m_{\mathrm{c}}, v_{\mathrm{c}}$ , and physical constants, as appropriate. Begin your

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

information.

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(iii) For a multistep derivation that includes the correct relationship between kinetic energy,

mass, and speed (i.e., $K=\frac{1}{2} m v^{2}$ )

Scoring Note: The minimum requirement to earn this point is to show an expression of

kinetic energy or change in energy that goes beyond the given equation on the reference

information. For example, substituting quantities from the problem into $K=\frac{1}{2} m v^{2}$ or

indicating the change in kinetic energy is $\Delta K=\frac{1}{2} m v_{f}^{2}-\frac{1}{2} m v_{i}^{2}$ earns the point.

For an indication that $m_{\mathrm{c}}$ and $\frac{6}{5} m_{\mathrm{c}}$ are the initial and final masses, respectively, of the

objects moving horizontally

For including an expression for $v_{f}$ consistent with part A (ii)

Scoring Note: A correct, isolated, final expression of $\Delta K=-\frac{1}{12} m_{\mathrm{c}} v_{\mathrm{c}}^{2}$ earns points

A3, A4, A6, and A7.

Example Response

$\begin{aligned} \Delta K & =K_{f}-K_{0} \\ K_{0} & =\frac{1}{2} m_{\mathrm{c}} v_{\mathrm{c}}^{2} \\ K_{f} & =\frac{1}{2}\left(\frac{6}{5} m_{\mathrm{c}}\right)\left(\frac{5}{6} v_{\mathrm{c}}\right)^{2}=\frac{5}{12} m_{\mathrm{c}} v_{\mathrm{c}}^{2} \\ \Delta K & =-\frac{1}{12} m_{\mathrm{c}} v_{\mathrm{c}}^{2} \end{aligned}$

(b)

Consider the case where a new block is dropped and collides with the top of the cart. The

new block slides along the cart during the collision but does not slide off the cart. The time

interval from when the new block collides with the cart and moves together with the cart

is $\Delta t$ . During $\Delta t$ there is a frictional force between the new block and the cart.

Indicate whether the x-component of the momentum of the new block-cart system increases,

decreases, or remains constant during $\Delta t$ .

Increases

Decreases

Remains constant

Justify your response.

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Covered syllabus points

4 Linear Momentum question 1

4 Linear Momentum question 2

Question (a)

Question (b)

4 Linear Momentum question 3

Question (a)

Question (i)

Question (ii)

Question (iii)

Question (b)