A block of mass m is thrown with a velocity $v_{0}$ towards a spring as shown in figure. Initially, the block is at a distance L from the spring and the spring is in its natural length. The coefficient of friction between the block and the ground is $μ = a x,$ where a is a constant and x is the distance travelled. Find value of $v_{0}$ such that the maximum compression in the spring is L.

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${[\frac{k L^{2}}{m} + 2 a L^{2} g]}^{1 / 2}$

${[\frac{k L^{2}}{m} + 4 a L^{2} g]}^{1 / 2}$

${[2 \frac{k L^{2}}{m} + a L^{2} g]}^{1 / 2}$

${[\frac{k L^{2}}{m} + a L^{2} g]}^{1 / 2}$

answer is C.

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Detailed Solution

Applying work-energy theorem $W_{m g} + W_{N} + W_{f r i c t i o n} + W_{s p r i n g} = Δ K$

$\Rightarrow 0 + 0 - \int_{0}^{2 L} (a x) m g d x + \frac{1}{2} k (0^{2} - L^{2}) = 0 - \frac{1}{2} m v_{0}^{2}$

$\Rightarrow v_{0} = \sqrt{\frac{k L^{2}}{m} + 4 a L^{2} g}$

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