AP Physics C: Mechanics 2.5 A Describe the Conditions Under Which a Systems Velocity Changes Questions

Blocks A and B of masses 2 m and m, respectively, are arranged in a setup consisting of a ramp that makes an angle $\theta$ with a smooth horizontal table and an ideal spring of spring constant k fixed to a wall, as shown. Block A is held at rest a distance D up the ramp, and Block B is at rest on the horizontal table. The coefficient of kinetic friction between Block A and the rough ramp is $\mu$ in the region of length D, and there is negligible friction between the blocks and the smooth table.

At time t=0, Block A is located at horizontal position x=0 and is released from rest. After the block is released, the following occurs.

- At time $t=t_{1}$, Block A has traveled a distance D down the ramp, has transitioned to the table, and is moving with speed v at $x=x_{1}$.

- At time $t=t_{2}$, Block A is at $x=x_{2}$ when it collides with and sticks to Block B.

- At time $t=t_{3}$, the combined blocks A and B are at $x=x_{3}$ when they collide with and stick to the spring in its equilibrium position.

- At time $t=t_{4}$, the combined blocks A and B are instantaneously at rest and the spring is compressed a distance $x_{\mathrm{c}}$ from its equilibrium position.

Use principles of forces to justify the graph drawn in part (b)(i) for the time interval $t=t_{3}$ to $t=t_{4}$. Explicitly reference features of the shape of the graph you drew in part (b)(i).

For times $t>t_{4}$, the two-block-spring system oscillates with period $T_{\mathrm{O}}$. The procedure is then repeated using a new ramp, where there is negligible friction between Block A and the ramp.