Question

Formula of range of a projectile thrown from top of inclined plain

Answer 1

1. The problem statement, all variables and given/known data A ball is thrown with initial speed V V up an inclined plane. The plane is inclined at an angle ϕ ϕ above the horizontal, and the ball's velocity is at an angle θ θ above the plane. Show that the ball lands a distance R = 2 V 2 sin θ cos ( θ + ϕ ) g cos 2 ϕ R=2V2sin⁡θcos⁡(θ+ϕ)gcos2⁡ϕ from its launch point. Show that for a given V V and ϕ ϕ , the maximum possible range up the inclined plane is R m a x = V 2 g ( 1 + sin ϕ ) Rmax=V2g(1+sin⁡ϕ) 2. Relevant equations F = ma 3. The attempt at a solution I calculated the distance traveled up the incline fine. However, I'm having trouble proving the second part. I'm guessing I'm supposed to maximize R with respect to theta, so from the equation above we have: d d θ sin θ cos ( θ + ϕ ) = 0 ddθsin⁡θcos⁡(θ+ϕ)=0 cos ( 2 θ + ϕ ) = 0 cos⁡(2θ+ϕ)=0 θ = n π 4 − ϕ 2 θ=nπ4−ϕ2 , with n = odd integer. Now plug this back into the original equation for R and I get