Question

A long block $A$ is at rest on a smooth horizontal surface. A small block $B$, whose mass is half of $A$, is placed on $A$ at one end and projected along $A$ with some velocity $u$. The coefficient of friction between the block is $\mu$.

• A. The blocks will reach a final common velocity $\frac{u}{3}$
• B. The work done against friction is two - thirds of the initial kinetic energy of B.
• C. Before the blocks reach a common velocity, the acceleration of A relative to B is $\frac{2}{3}\mu g$
• D. Before the blocks reach a common velocity the acceleration of A relative to B is $\frac{3}{2}\mu g$

Answer 1

Answer: (A), (B), (D)

Before the blocks reach a common velocity the acceleration of A relative to B is $\frac{3}{2}\mu g$

Let $v$ be the final common velocity,
Using conservation of momentum,
$mu=(m+2m)v$,
$v=\frac{u}{3}$
Work done by friction is change in loss of kinetic energy, so
$\frac{1}{2}m{u}^2-\frac{1}{2}\left(3m\right)\frac{u^2}{9}=\frac{1}{2}m{u}^2[1-\frac{1}{3}]=\frac{2}{3}\times \frac{1}{2}m{u}^2$.
The work done agianst friction is two - thirds of the initial kinetic energy of B

The force of friction between the blocks is $\mu mg.$
Acceleration of A(to the right) $=a_1=\frac{\mu mg}{2m}=\frac{\mu g}{2}$
Acceleration of B (to the left)$=a_2=\frac{\mu mg}{m}=\mu g$
Acceleration of A relative to B$=a_1-(-a_2)=\frac{3}{2}\mu g.$