A block of mass  $M=5\mathrm{kg}$ is moving on a  horizontal table and the coefficient of friction is $\mathrm{\mu }=0.4$A clay ball of mass m = 1 kg is dropped on the block, hitting it with a vertical velocity of $u=10m/s$ At the instant of hit, the block was having a horizontal velocity of $\mathrm{v}=2\mathrm{m}/\mathrm{s}$ After an interval of  $∆\mathrm{t}$, another similar clay ball hits the block and the system comes to rest immediately after the hit. Assume that the clay balls stick to the block and collision is momentary. Find $∆\mathrm{t}$  .Take g $=10\mathrm{m}/{\mathrm{s}}^2$ 

During the interaction period of the clay ball and the block, the vertical impulse (due to normal force) applied by ground is 
${\mathrm{J}}_{\mathrm{v}}=\int \mathrm{Ndt}=$ change in momentum in
vertical direction $1=1\times 10=10\mathrm{kgm}/\mathrm{s}\dots \dots \dots \dots ..\left(\mathrm{i}\right)$
Horizontal impulse of ground friction during the same period is 
${\mathrm{J}}_{\mathrm{H}}=\int \mathrm{\mu Ndt}=0.4\times 10=4\mathrm{kgm}/\mathrm{s}$    ……..(ii)
 

Velocity (V₁) of the (block + ball) system just after the impact is given by 
$$\begin{gathered} (\mathrm{M}+\mathrm{m}){\mathrm{V}}_1=\mathrm{MV}-{\mathrm{J}}_{\mathrm{H}} \\ {\mathrm{V}}_1=\frac{5\times 2-4}{6}=1\mathrm{m}/\mathrm{s} \end{gathered}$$
Let the velocity of (M + m) be reduced to V₂ [due to friction] in interval $∆$t

At this point another clay ball hits the block.
${\mathrm{J}}_{\mathrm{V}}\text{ and }{\mathrm{J}}_{\mathrm{H}}$H given by (i) and (ii) remain same for the second impact 
$$\begin{gathered} \therefore 0=(\mathrm{M}+\mathrm{m}){\mathrm{V}}_2-{\mathrm{J}}_{\mathrm{H}} \\ \therefore {\mathrm{V}}_2={\mathrm{V}}_1-\mathrm{a\Delta t} \end{gathered}$$
During the interval between two impact, friction causes a retardation of

$$\begin{gathered} \mathrm{a}=\mathrm{\mu g}=4\mathrm{m}/{\mathrm{s}}^2 \\ \therefore {\mathrm{V}}_2={\mathrm{V}}_1-\mathrm{a\Delta t} \\ \Rightarrow \mathrm{\Delta t}=\frac{1-\frac{2}{3}}{4}=\frac{1}{12}\mathrm{se}c \end{gathered}$$