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A massless spring of force constant $k$ has two ends $P$ and $Q$ as shown in the figure. End $Q$ is being pulled by a person $B$ with a constant speed $' u'$ and end $P$ is being pushed by a person $A$ with constant acceleration $' a'$ towards right starting from rest. Both starts simultaneously. Then pick out the correct statements of the following at the instant when $B$ appears to be at rest with respect to $A$

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elastic potential energy stored in the spring is $\frac{k u^{4}}{8 a^{2}}$

work done by $A$ is negative

work done by $B = \frac{k u^{4}}{3 a^{2}}$

work done by $A = - \frac{5 k u^{4}}{24 a^{2}}$

answer is A, B, C, D.

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Detailed Solution

Work done by $B = \int_{0}^{u / a} k (u t - \frac{1}{2} a t^{2}) u d t = \frac{k u^{4}}{3 a^{2}}$

Work done by $A = - \int_{0}^{u / a} k (u t - \frac{1}{2} a t^{2}) a t d t = - \frac{5 k u^{4}}{24 a^{2}}$

Energy stored in spring $= \frac{1}{2} k {(\frac{u^{2}}{2 a})}^{2} = \frac{k u^{4}}{8 a^{2}}$

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