A projectile of mass, $,$ collides with a pendulum bob of mass, $,$ such that after the collision the projectile has half its original velocity in approximately the same direction. The collision will cause the pendulum bob to have a speed such that it begins to swing upward. $($a$)$ Use conservation of momentum to get an equation relating the initial speed of the projectile to the speed of the bob after the collsion. $($b$)$ Use conservation of energy to find an equation for how fast the bob must go to swing around in a full circle on a light string. $($c$)$ Combine $($a$)$ and $($b$)$ to get an equation for the initial speed of the projectile needed to knock the pendulum bob on the light string in a full circle.