Ball fired from cannon homework

[zippyjibby]

Homework Statement

A cannon, when aimed vertically, is observed to fire a ball to a maximum height of L. Another ball is then fired with the same speed, but with the cannon aimed up along an incline of length L, inclined at an angle θ. What should θ be so that the ball travels the largest horizontal distance d by the time it returns to the height of the top of the plane?

2. The attempt at a solution
My attempt is a bit messy, as you can see below, but I tried to find an equation to describe the initial velocity at the base of the ramp, and an equation to describe the velocity as it leaves the ramp. Then I plugged that into basic projectile motion equations, and got d in terms of constants and θ. I then took the deriviative to find the optimal angle, but I could not figure out the equation. The homework is due Friday (9/18/2015) at 5pm, and I have not been able to figure it out.

 

[Ray Vickson]

Homework Statement

A cannon, when aimed vertically, is observed to fire a ball to a maximum height of L. Another ball is then fired with the same speed, but with the cannon aimed up along an incline of length L, inclined at an angle θ. What should θ be so that the ball travels the largest horizontal distance d by the time it returns to the height of the top of the plane?

2. The attempt at a solution
My attempt is a bit messy, as you can see below, but I tried to find an equation to describe the initial velocity at the base of the ramp, and an equation to describe the velocity as it leaves the ramp. Then I plugged that into basic projectile motion equations, and got d in terms of constants and θ. I then took the deriviative to find the optimal angle, but I could not figure out the equation. The homework is due Friday (9/18/2015) at 5pm, and I have not been able to figure it out. View attachment 88781View attachment 88782View attachment 88783
View attachment 88784

I, for one, seldom attached photos of rough work, and never those that are sideways. I will look at it if you type it out.