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**1. Problem**

I have a horizontal cylinder with a ball in it. What must the initial pressure be to launch the ball at Z velocity?

**2. Attempt**

P = F/A

A = Surface Area(Cylinder) - 1/2 Surface Area (Ball) = (2πrh)

F = m*a

U = - ∫ F dx

U = 1/2 mv^2 at x = h

∫ a dx = -1/2mv^2 / m at x = h

Z = -1/2v^2

P = F/A

A = Surface Area(Cylinder) - 1/2 Surface Area (Ball) = (2πrh)

F = m*a

U = - ∫ F dx

U = 1/2 mv^2 at x = h

∫ a dx = -1/2mv^2 / m at x = h

Z = -1/2v^2

This is where I am stuck. How can the final velocity be 1/2 final velocity squared? Any help would be greatly appreciated