Projectile motion of an arrow to an apple

[ornitho]

im having a very difficult time with this question:

William Tell is said to have shot an apple off his son's head with an arrow. If the arrow was shot with an initial speed of 55m/s and the boy was 25m away, at what launch angle did William aim the arrow? The arrow and apple are initially at the same height above the ground.

 

[courtrigrad]

$$x = v_{0}\cos \theta \; t$$
$$y = v_{0}\sin \theta\; t -\frac{1}{2}gt^{2}$$

$$v_{0} = 55$$So you have:

$$25 = 55\cos \theta \; t$$
$$0 = 55\sin \theta\; t - 5t^{2}$$

 

[ornitho]

I do not have time value though :(

 

[ornitho]

y = 0 wouldn't it, as it is returning to the initial y position, but i still do not understand how I am supposed to find t

 

[courtrigrad]

The maximum altitude occurs at $$t = \frac{v_{0}\sin\theta}{g}$$.

The maximum altitude is: $$\frac{v_{0}^{2}}{2g}\sin^{2}\theta$$

 

[physics girl phd]

y = 0 wouldn't it, as it is returning to the initial y position, but i still do not understand how I am supposed to find t

That's right! now if you understand courtrigrad's earlier equations, for velocity and position, you have two equations, and two unknowns (t and the angle, theta). You can then solve the set of equations for both unknowns.