Answer to Question #86720 in Classical Mechanics for Shahi

Question

At t= 0, a block of mass m is projected with speed v on a rough long horizontal plank (of mass 3m) placed on smooth ground, as shown. Work done by friction on the block between t= 0 and t= infinity equals

Answer 1

Using momentum conservation law:

$$p_{initial}=p_{final} \rightarrow mv=(m+3m)v_f$$

$$v_f=\frac{mv}{4m}=0.25v$$

where, v_f - speed of two blocks at t=infinity

Total work done by friction force equals to the loss in kinetic energy:

$$A=\Delta E=\frac{mv^2}{2}-\frac{(m+3m)v_f^2}{2}$$

$$A=\frac{mv^2}{2}-\frac{mv^2}{8}$$

$$A=\frac{3mv^2}{8}$$

So, total work done by friction force is

$$A=3mv^2/8$$