Barbarians hiding in a forest at attacking your village with a catapult. You can

Question

Barbarians hiding in a forest at attacking your village with a catapult. You can’t see them so you are trying to locate them using nothing but a very accurate clock. The stone projectile is first seen as they emerge from the forest at the top of the 8 meters tall trees (point A). Your measurements show that it takes the stone 2.83 seconds to reach the edge of the forest 80 meters from your village (point B). The stone arrives at your village 4.00 seconds after that (point C).

For both parts, assume that the ground is flat everywhere and that there is no air resistance.

https://gyazo.com/ea389a916356c2ba9a9a0533ee967248 (Link to image)

How far is the barbarian’s catapult from your village?

You set up a counter-catapult 30 meters from the edge of the forest. It launches stones at a speed of 33 m/s. You also want your stones to come down on the barbarians’ catapult at the steepest angle possible to avoid hitting the trees.

https://gyazo.com/4691a92ba9aad5e0cd8ca32e6e3c40a4 (Link to image)

What angle should you launch your stones to hit the barbarian catapult?

Explanation / Answer

Time taken to go from A to B, t1 = 2.83 s
Time taken to go from B to C, t2 =4 s
Assume barbarians shot the projectile wil velocity v and angle p
Assume the horizontal distance from catapult to A be x, A to B be y, and B to C be z.
so, vcosp = y/2.83 = z/4
but z = 80 m,
so, y = 56.6m
and vcosp = 20 ---- (1)
now, take vertical motion from A to C. Compare that with the ball being thrown up from a hieght of 8m
initial velocity = u'
using s = ut - at2/2
-8 = u' * 6.83 - 9.8*6.832/2
or u' = 32.39m/s
If initial velocity at the time of launch is considered, v'sinp
2*9.8*8 = ( v'sinp )2 - 32.39*32.39
v'sinp = 34.72 -- (2)
square and add 1 and 2 to get v = =40.06m/s and p = 60.077 degrees

Hence, total range R = v2sin2p/g = 40.06*40.06*sin120.14/9.8 = 141m
Distance from edge of forest to barbarian catapule = 141-80 = 61m
Distance of catapult from forest = 30m
Intended range = 91m, initial v = 33m/s, angle = p
so  R = v2sin2p/g
so 91 = 33*33sin2p/9.8
or p = 62.5 degrees

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