A block of mass $m$ is pushed against a spring of constant $k$ by a distance $x$ on a rough horizontal surface with coefficient of friction $\mu$. When released, the block travels a distance $d$ before coming to rest.

Solution

1. The initial elastic potential energy stored in the spring is $U=\frac{1}{2}k{x}^2$.
2. The work done by friction over the distance $d$ is $W_f=\mu mgd$.
3. By the work-energy theorem, the initial energy equals the work dissipated by friction: $\frac{1}{2}k{x}^2=\mu mgd$.
4. Solving for the distance $d$, we obtain $d=\frac{k{x}^2}{2\mu mg}$.
5. Hence the answer is (A).