energy of the moving coils of a spring, but let's try to get a reasonable approximation to this. Consider a spring of mass $M$, equilibrium length $L_{0},$ and force constant $k$. The work done to stretch or compress the spring by a distance $L$ is $\frac{1}{2} k X^{2},$ where $X=L-L_{0}$. Consider a spring, as described above, that has one end fixed and the other end moving with speed $v$. Assume that the speed of points along the length of the spring varies linearly with distance $l$ from the fixed end. Assume also that the mass $M$ of the spring is distributed uniformly along the length of the spring. (a) Calculate the kinetic energy of the spring in terms of $M$ and $v$. (Hint: Divide the spring into pieces of length $d l ;$ find the speed of each piece in terms of $l, v,$ and $L ;$ find the mass of each piece in terms of $d l, M,$ and $L ;$ and integrate from 0 to $L .$ The result is $n o t$ $\frac{1}{2} M v^{2},$ since not all of the spring moves with the same speed.) In a spring gun, a spring of mass $0.243 \mathrm{~kg}$ and force constant $3200 \mathrm{~N} / \mathrm{m}$ is compressed $2.50 \mathrm{~cm}$ from its unstretched length. When the trigger is pulled, the spring pushes horizontally on a $0.053 \mathrm{~kg}$ ball. The work done by friction is negligible. Calculate the ball's speed when the spring reaches its uncompressed length (b) ignoring the mass of the spring and (c) including, using the results of part (a), the mass of the spring. (d) In part (c), what is the final kinetic energy of the ball and of the spring?