AP Physics C: Mechanics 4.2 A Describe the Impulse Delivered to an Object or System Questions

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.

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.