HELP QUICK! Physics Problem, Projectiles!?

Question

An archer is standing inside a building whose ceiling is 12 m high. An arrow is shot from ground level at an initial speed of 62 m/s. Calculate the angle of firing (above the horizontal) that gives the greatest range inside the building. Neglect friction.

Thanks for any help you can give me, I'm stumped.

Answer 1

For this, we need to recognize that (1) this travels in a parabolic path, (2) the initial velocity can be borken down into x- and y-component velocities, and (3) at the top the velocity is 0. So, we can use vf^2 = vi^2 + 2ad to figure this one out.

The initial velocity is 62 m/s, but we can break this down into 62sinx and 62cosx for the horizontal and vertical velocities. We only need the vertical velocity, so we can say that the arrow is moving initially at 62cosx m/s up at the time it is shot.

At the top of the arrow's flight, it should reach 12 m, and the verticla velocity is 0. Given all of this (and that gravity acts against the arrow), we can plug in to the equation and solve for x:

vf^2 = vi^2 + 2ad
0 = (62cosx m/s)^2 + 2(-9.81 m/s^2)(12 m)
0 = 3844(cosx)^2 m^2/s^2 - 235.44 m^2/s^2
235.44 m^2/s^2 = 3844(cosx)^2 m^2/s^2
0.9388 = (cosx)^2
cosx = 0.9689
x = 14.33

So, shooting up at 14.33 degrees will give you the most range for this shot.