Initial pressure to launch ball at Velocity Z

[scain6043]

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

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

 

[Buzz Bloom]

Hi scain:

I think (1) the problem statement is incomplete, and (2) you have misunderstood several things.

1. The amount of velocity from the initial pressure depends on the length of the cylinder. What is it? Also you to need to state more clearly the initial configuration.
(a) I am guessing the cylinder as having a flat closed end and an open end from which the moving ball will emerge.
(b) I am guessing the ball rests touching the closed end of the cylinder.
(c) I am guessing a valve allows gas to enter the space between the ball and the closed end.
(d) I am guessing the ball in constrained from moving until the pressure behind the ball reaches some target pressure, and then the constraint is removed.
Are all these guesses correct, or if not, what is the configuration?

2. The area of interest for calculating the force on the ball is the half surface of the ball near the closed end of the cylinder. You need to integrate the component of force, from the pressure on each dA of the surface, in the direction of the axis of the cylinder toward the open end. You need to calculate how the pressure changes as the ball moves. You need to calculate the changing force on the ball as the ball moves until its center passes the open end of the cylinder.

Hope this helps.

Regards,
Buzz