AP Physics C: Mechanics 4.2 Change in Momentum and Impulse Questions

Analyse impulse as both a net-force time integral and a vector momentum change, using functions, graph areas, rebounds, collision duration, and inverse calculations.

Exam points

calculate vector impulse from constant force, a time integral, or force-time graph area
apply the impulse-momentum theorem with correct initial and final momentum signs
analyse rebounds and collisions where momentum direction changes during the interaction
relate a fixed impulse to average force, contact time, mass, and velocity change
combine multiple force intervals or impulses to find the net change in system momentum

Covered syllabus points

4.2.A—Describe the impulse delivered to an object or system 4.2.B—Describe the relationship between the impulse exerted on an object or system and the change in momentum of the…

[Maximum number: 5]

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

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.

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.

(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

[Maximum number: 1]

Scientists have created a new type of lightweight foam and are performing experiments to investigate the properties of the foam. The mass of Cart A is 1000 kg and the mass of Cart B is 2000 kg. A piece of foam with negligible mass is attached to the front of Cart A, as shown. Cart A moves with a constant speed toward Cart B, which is initially at rest. At time $t=0 \mathrm{~s}$ , the foam connected to Cart A makes contact with Cart B. The foam remains in contact with Cart B for 0.5 s, after which the carts separate and both carts move with constant velocities.

Figure for Question 2 — AP Physics C: Mechanics

What is the relationship between the magnitude of the average net force $F_{1}$ exerted on Cart B for the collision with the foam and the magnitude of the average net force $F_{2}$ exerted on Cart B for the collision without the foam? $F_{1}>F_{2}$ $F_{1}<F_{2}$ $F_{1}=F_{2}$
Justify your answer.

For selecting $F_{1}<F_{2}$ with an attempt at a relevant justification

1 point

For indicating that the impulse or change in momentum of each cart in both collisions is the
same

1 point

For indicating that decreasing the time of collision means the average force must be greater

1 point

Example Solution
Since the initial and final velocities are the same for both collisions, $\Delta p$ is the same for both collisions;
as a result, the impulse is the same for both collisions. So, if $\Delta t$ is smaller, $F_{\text {avg }}$ is larger.

Total for part (c)

for question 1

15 points

All question bank results loaded

Covered syllabus points

Question 1

Question 2