Chapter 7, Problem 29P

To determine

The expression of distance $s$ in terms of relevant physical quantities. Given that the spring of spring constant $k$ is compressed by distance $x$ to shoot a steel ball of mass $m$ into a sponge of mass $M$. The ball and sponge move though a distance $s$ after collision along a rough surface as shown in the figure. The coefficient of friction between sponge and surface be $\mu$.

Devin derived the following expression,

$s=\frac{km{x}^2}{2{\left(m+M\right)}^{2}g\mu }$

Without deriving it, the equation is reasonable. Explain.