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 a velocity $v$. The coefficient of friction is $\mu$.Then,

• A. the block will reach a final common velocity $u/3$.
• B. the work done against friction is $2/3$ of initial $KE$ of $B$.
• C. before the blocks reach a common velocity, the acceleration of $A$ relative to $B$ is $2/3\mu g$.
• D. before the blocks reach a common velocity the acceleration of $A$ relative to $B$ is $3/2\mu g$.

Answer 1

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

The correct options are
A the block will reach a final common velocity $u/3$.
B the work done against friction is $2/3$ of initial $KE$ of $B$.
D before the blocks reach a common velocity the acceleration of $A$ relative to $B$ is $3/2\mu g$.

initial Momentum = Final Momentum (as no external force)
$mu=(m+2m)v$
$\Rightarrow v=\frac{u}{3}$
Intial KE $K_i=\frac{1}{2}m{u}^2$
Final KE: $K_f=\frac{1}{2}(m+2m)(\frac{u}{3}{)}^2=(\frac{1}{6})m{u}^2$
Therefore work done against friction $W=(\frac{1}{2}-\frac{1}{6})m{u}^2=\frac{1}{3}m{u}^2=\frac{2}{3}{K}_i$
Force on B $F_B=-\mu mg$ ; Force on A $F_A=\mu mg$
Acceleration of B: $a_B=-\mu g$;
Acceleration of A: $a_A=\frac{\mu g}{2}$;
Relative acceleration between A and B: $a_{AB}=\mu g+\frac{\mu g}{2}=\frac{3}{2}\mu g$