The maximum compression produced in a spring of force constant k, when a block of weight W moving with velocity v on a frictionless horizontal surface compresses it, shall be:
\({\rm{A) v}}\dfrac{{\sqrt {\rm{w}} }}{{\sqrt {\rm{k}} }}\\{\rm{B) v}}\dfrac{{\sqrt {\rm{k}} }}{{\sqrt {\rm{w}} }}\\{\rm{C) v}}\dfrac{{\sqrt {\rm{w}} }}{{\sqrt {{\rm{k}}{\rm{.g}}} }}\\{\rm{D) v}}\dfrac{{\sqrt {{\rm{k}}{\rm{.g}}} }}{{\sqrt {\rm{w}} }}\)​​

Hint: When the spring is compressed the kinetic energy increases and the elastic potential energy decreases. The total amount of these two forms of mechanical energy remains constant. Mechanical energy is being converted from potential form to kinetic form, thus saving the total amount.

Complete step by step solution:
\({\rm{so}}\dfrac{{\rm{1}}}{{\rm{2}}}{\rm{m}}{{\rm{v}}^{\rm{2}}}{\rm{ = }}\dfrac{{\rm{1}}}{{\rm{2}}}{\rm{k}}{{\rm{x}}^{\rm{2}}}\\x = \sqrt {\dfrac{{m{v^2}}}{k}} \)
From conservation of energy
KE lost by the block = PE gained by the spring
\(\Rightarrow \dfrac{{\rm{1}}}{{\rm{2}}}{\rm{m}}{{\rm{v}}^{\rm{2}}}{\rm{ = }}\dfrac{{\rm{1}}}{{\rm{2}}}{\rm{k}}{{\rm{x}}^{\rm{2}}}\\ \Rightarrow x = \sqrt {\dfrac{{{\rm{m}}{{\rm{v}}^{\rm{2}}}}}{{\rm{k}}}} \\ \Rightarrow \sqrt {\dfrac{{{\rm{W}}{{\rm{v}}^{\rm{2}}}}}{{{\rm{gk}}}}} \\\therefore {\rm{\text{maximum compression} = v}}\sqrt {\dfrac{{\rm{W}}}{{{\rm{kg}}}}} \)
So option C is correct.

Additional Information: Compression springs are open-coil helical springs wound or built to resist compression along the wind axis. Helical compression is the most common metal spring configuration. These coil springs can operate independently, although often assembled on a guide rod or fitted inside a hole.
When we place a load on a compression coil spring, make it shorter, push it back against the load and try to get it back to its original length. Compression springs offer linear compressive force (push) resistance and are in fact one of the most effective energy storage devices.
Compression springs are found in a wide range of applications, from automotive engines and large stamping presses to lawn mowers to major appliances and medical devices, cell phones, electronics, and sensitive instrumentation devices. The most basic installation anywhere requires a push button. Conical type springs are typically used in applications required for low concrete height and increased resistance.

Note: When a spring is compressed, the spring is worked by external agent forces. Suppose this task is accomplished by a moving object that initially moves to its equilibrium position [position A] in a spring so that the moving object loses momentum (eventually comes to rest) it acts in the spring to contract [position B].
As the spring shrinks and the mass slows down, its kinetic energy changes to elastic potential energy. As this changes, the total amount of mechanical energy is conserved.