AP Physics C Mechanics Unit 4: Linear Momentum

Analyze interactions with momentum and impulse, select appropriate systems, and distinguish elastic from inelastic collisions using conservation laws.

[Maximum number: 10]

Two blocks, 1 and 2, slide toward each other on a horizontal surface. Block 1 has mass m

and slides in the +x-direction with constant speed $2 v_{0}$ . Block 2 has mass 6 m and slides in the

-x-direction with constant speed $v_{0}$ , as shown in Figure 1. The blocks then collide and stick

together. The collision occurs from time t=0 to $t=t_{c}$ . After the collision, where $t>t_{c}$ , the blocks

move together with the same constant speed.

Figure 1

(a)

The diagrams in Figure 2 can be used to represent the momentum of blocks 1 and 2 before

and after the collision. The momentum vector diagram for Block 1 before the collision is

shown.

[ 7 ]

(i)

Draw arrows on the grids to represent the momentum vectors of Block 2 before the

collision and the two-block system before and after the collision.

- Arrows should start at the zero-momentum line.

- The length of the arrows should be proportional to the relative magnitudes of the

vectors.

- Represent an arrow of zero length by drawing a dot at zero.

Figure 2

[ 2 ]

(ii)

During the time interval $0 \leq t \leq t_{c}$ , a force F is exerted on Block 2 by Block 1 along the

x-direction as a function of t that is modeled by $F(t)=F_{\max } \sin (A t)$ , where A is a positive

constant and $F_{\text {max }}$ is the magnitude of the maximum force exerted on Block 2 by Block 1

during the collision.

Derive an expression for $F_{\text {max }}$ . Express your answer in terms of $m, v_{0}, A, t_{c}$ , and physical

constants, as appropriate. Begin your derivation by writing a fundamental physics

principle or an equation from the reference information.

[ 5 ]

(ii) For a multistep derivation that includes the integral form of impulse or the differential form of Newton's second law & Point A3 \\ \hline & For indicating one of the following:

- The final speed of Block 1 or Block 2 is $\frac{4}{7} v_{0}$ .
- The change in the momentum of Block 2 is $+\frac{18}{7} m v_{0}$ .
- The change in the momentum of Block 1 is $-\frac{18}{7} m v_{0}$ .
- The change in the velocity of Block 2 is $+\frac{3}{7} v_{0}$ .
- The change in the velocity of Block 1 is $-\frac{18}{7} v_{0}$ .

& Point A4 \\ \hline & For substituting the given expression for F(t) into an expression for the impulse exerted on either block or the differential form of Newton's second law & Point A5 \\ \hline & For an integral with appropriate limits or a constant of integration & Point A6 \\ \hline & For a correct expression for $F_{\text {max }}$ in terms of given quantities & Point A7 \\ \hline \end{tabular}

Example Responses

	$\vec{J}=\int_{t_{1}}^{t_{2}} \vec{F}_{\text {net }}(t) d t=\Delta \vec{p} \quad \text { OR } \quad \begin{aligned} & \vec{F}_{\text {net }}=\frac{d \vec{p}}{d t} & \int \vec{F}_{\text {net }} d t=\Delta \vec{p} \end{aligned}$
		$\sum \vec{p}_{0}=\sum \vec{p}_{f}$ $(m)\left(2 v_{0}\right)+(6 m)\left(-v_{0}\right)=(m+6 m)\left(v_{f}\right)$ $v_{f}=-\frac{4}{7} v_{0}$ $\sum \vec{p}_{0}=\sum \vec{p}_{f}$ $(m)\left(2 v_{0}\right)+(6 m)\left(-v_{0}\right)=(m+6 m)\left(v_{f}\right)$ $v_{f}=-\frac{4}{7} v_{0}$

Official scoring-guideline example response

(b)

Consider a new scenario where Block 1 initially slides in the +x-direction with a new

constant speed $v_{1}$ and Block 2 again initially slides in the -x-direction with constant speed $v_{0}$ .

The blocks collide and stick together. In this new scenario, the two-block system has constant

speed $v_{0}$ after the collision.

Derive an expression for $v_{1}$ in terms of $v_{0}$ . Begin your derivation by writing a fundamental

physics principle or an equation from the reference information.

[ 3 ]

B	For a multistep derivation that includes conservation of momentum	Point B1
		For indicating that the magnitude of the final momentum of the two-block system is $7 m v_{0}$	Point B2
		For a correct expression for $v_{1}$ in terms of $v_{0}$	Point B3
		Example Response $\begin{aligned} & \sum \vec{p}_{0}=\sum \vec{p}_{f} & m v_{1}+6 m\left(-v_{0}\right)=7 m\left(v_{0}\right) & m v_{1}=13 m v_{0} & v_{1}=13 v_{0} \end{aligned}$

[Maximum number: 2]

A horizontal circular platform with rotational inertia $I_{P}$ rotates freely without friction on a vertical axis. A small motor-driven wheel that is used to rotate the platform is mounted under the platform and touches it. The wheel has radius r and touches the platform a distance D from the vertical axis of the platform, as shown above. The platform starts at rest, and the wheel exerts a constant horizontal force of magnitude F tangent to the wheel until the platform reaches an angular speed $\omega_{P}$ after time $\Delta t$ . During time $\Delta t$ , the wheel stays in contact with the platform without slipping.

The kinetic energy of the spinning platform before the object is dropped on it is $K_{i}$ . The total kinetic energy of the platform-object system when it reaches angular speed $\omega_{f}$ is $K_{f}$ . Which of the following expressions is true? $K_{f}<K_{i}$ $K_{f}=K_{i}$ $K_{f}>K_{i}$
Justify your answer.

LO CON-5.D.c, SP 7.A, 7.C 2 points

The kinetic energy of the spinning platform before the object is dropped on it is $K_{i}$ . The total kinetic energy of the platform-object system when it reaches angular speed $\omega_{f}$ is $K_{f}$ . Which of the following expressions is true?

K_{f}<K_{i}

\ldots K_{f}=K_{i}

\ldots K_{f}>K_{i}

Justify your answer.

For selecting $K_{f}<K_{i}$ with an attempt at a relevant justification

1 point

For a correct justification

1 point

Example: Because the two disks will be rotating with the same final angular speed, this
is an inelastic collision, and kinetic energy will be lost during the collision.

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Covered syllabus points

4 Linear Momentum question 1

Question (a)

Question (i)

Question (ii)

Question (b)

4 Linear Momentum question 2