a) 20 m/s b) 10 m/s c) 30 m/s d)40 m/s

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Maximum change of slipping can occur when the spring is compressed maximum.

Force provided by spring $=kx_m$

Acceleration of system of two mass $=\large\frac{kx_m}{M+m}$

For not slipping

Frictional force between two blocks must be greater than $ma$

$\mu mg >ma$

$\mu mg >\large\frac{kx_mm}{M+m}$-----(1)

Using work energy theorem for system

$\large\frac{1}{2}$$(M+m)v^2=\large\frac{1}{2}$$kx_m^2$

$x_m=\sqrt {\large\frac{(M+m)v^2}{k}}$-----(2)

Substituting for $x_m$ in (1) we get, limiting values

$\mu mg=\large\frac{kV \sqrt {\Large\frac{M+m}{k}}m}{M+m}$

Therefore $v=\mu g\sqrt {\large\frac{M+m}{k}}$

$\qquad=.2 \times 10 \sqrt {\large\frac{3+1}{400}}$

Pulling values for $\mu,g,M\; and\;m$

we get $v=20\;m/s$

Hence a is the correct answer.

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