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System:  Projectile as [point particle] plus the gun as a rigid body of infinite mass.

Interactions:  The [conservative|conservative force] spring interaction between the gun and the projectile will give rise to [elastic potential energy].

Model: [Mechanical Energy and Non-Conservative Work].

Approach:  We will ignore friction and other [non-conservative] interactions, which means that the [mechanical energy] of the system will be constant.  We will further make the usual assumption that the projectile stops interacting with the spring when the spring returns to its equilibrium position (the projectile has essentially been "fired" at that point).  Thus, appropriate [initial-state final-state diagram] and [energy bar graphs|energy bar graph] are:

|!springgun1.png!|!springgun2.png!|
||Initial||Final||

We can now express the fact that the mechanical energy is constant through the Law of Change:

{latex}\begin{large}\[ E_{i} = U_{i} = \frac{1}{2}kx_{i}^{2} = E_{f} = K_{f} = \frac{1}{2}mv_{f}^{2}\]\end{large}{latex}