Mickey, a daredevil mouse of mass m , m, is attempting to become the world's first "mouse cannonball." He is loaded into a spring-powered gun pointing up at some angle, and is shot into the air. The gun's spring has force constant k k and is initially compressed a distance X X from its relaxed position. If Mickey has a constant horizontal speed V V while he is flying through the air, how high above his initial location in the gun does Mickey soar? Find an expression for Mickey's maximum height h m hm in terms of m , m, V , V, X , X, k , k, and g .

  h = v₀² / 2g ,      h = k/4g     x²

Explanation:

In this exercise we can use the law of conservation of energy at two points, the lowest, before the shot and the highest point that the mouse reaches

Starting point. Lower compressed spring

              Em₀ = K = ½ m v²

Final point. Highest on the path

              = U = mg h

As or no friction the energy is conserved  

              Em₀ =  Em_{f}

              ½ m v₀²² = m g h

             h = v₀² / 2g

We can also use as initial energy the energy stored in the spring that will later be transferred to the mouse

                  ½ k x² = 2 g h

                  h = k/4g     x²