# Projectile Motion with Relative Motion (w/o Solver)

1. Apr 9, 2016

### bartersnarter

1. The problem statement, all variables and given/known data

2. Relevant equations
vx = 70*cos(α) + 8.333 (30 km/h = 8.333 m/s)
vy = 70*sin(α)
vx*t = 350
vy*t = 60
-(1/2)*g*t^2 + vy*t + 2.5 = 60

3. The attempt at a solution
This problem is conceptually very simple for me, but I can't solve it without using a solver like Wolfram Alpha. First, I simply rearranged vx*t = 350 into t = 350/vx = 350/(70*cos(α) + 8.333). Then, I plugged this t into the -(1/2)*g*t^2 + vy*t + 2.5 = 60 equation and I got an equation with only the angle α as the unknown. After moving the equation around so that there are no denominators, I get:
600825.5 + 204167*sin(α) + 1715000*sin(α)*cos(α) = 281750*cos2(α) + 67083*cos(α) + 3993

Is there a way to solve this equation without using a solver? Is there another method entirely which I'm missing? I have the right answers, which are α = 31.32° and 74.28°.

2. Apr 10, 2016

### haruspex

You could turn it into a quartic in cos(α), but you would not be that much better off.