Question

scenario
in the diagram shown to the right a block of mass m has taken a quick hit from a bat. after the strike, its front end is at position x = 0 at time t = 0 and it is moving to the right with initial speed v0. the block slides on a rough surface and is also connected to a hanging mass object of mass m by a string that passes over an ideal pulley. the front end of the block reaches position x = d at time t = t1, the instant that the block comes to rest. the block then returns to position x = 0 at time t = t2, having a left - ward speed v2

using representations
part a: the dots below represent the block on the table during the interval 0 < t < t1 and t1 < t < t2. draw free - body diagrams showing and labeling the forces (not components) exerted on the block during each of those intervals. draw the relative lengths of all vectors to reflect the relative magnitudes of all the forces. each force should be a single arrow that originates on the dot.
forces during 0 < t < t1
forces during t1 < t < t2

analyze data
part b: is the magnitude of the blocks acceleration greater before the block reaches x = d or after? explain your reasoning in terms of the forces that you drew in the above diagrams.

Explanation:

Step1: Analyze forces for $0 < t < t_1$

The block has a right - ward initial velocity. Forces acting on it are: gravitational force $F_g = Mg$ downwards, normal force $N$ upwards (where $N = Mg$ as there is no vertical acceleration), tension force $T$ to the right from the string connected to the hanging mass, and kinetic friction force $f_k=\mu_k N=\mu_k Mg$ to the left. On the free - body diagram, draw a dot to represent the block. Draw a long arrow downwards for $F_g$, an equal length arrow upwards for $N$, a shorter arrow to the right for $T$ and a shorter arrow to the left for $f_k$.

Step2: Analyze forces for $t_1 < t < t_2$

The block is moving to the left. Forces are: gravitational force $F_g = Mg$ downwards, normal force $N$ upwards ($N = Mg$), tension force $T$ to the right from the string connected to the hanging mass, and kinetic friction force $f_k=\mu_k N=\mu_k Mg$ to the right (since the block is moving left). On the free - body diagram, draw a dot for the block. Draw a long arrow downwards for $F_g$, an equal length arrow upwards for $N$, an arrow to the right for $T$ and an arrow to the right for $f_k$.

Step3: Analyze acceleration for part B

According to Newton's second law $F_{net}=ma$. Before reaching $x = D$ (for $0 < t < t_1$), $F_{net1}=f_k - T=Ma_1$. After reaching $x = D$ (for $t_1 < t < t_2$), $F_{net2}=f_k + T=Ma_2$. Since $F_{net2}>F_{net1}$ and $m = M$ is constant, from $a=\frac{F_{net}}{M}$, the magnitude of the block's acceleration is greater after the block reaches $x = D$.

Answer:

The magnitude of the block's acceleration is greater after the block reaches $x = D$.