A projectile, fired with unknown velocity, lands 22s later 459 vertically on the

Question

A projectile, fired with unknown velocity, lands 22s later 459 vertically on the side of a hill that is 3140 m away horizontally away from its starting point. (Air resistance is negligible and g = 9.81m/s^2)

a) What is the vertical component, voy, of its initial velocity?

b) What is the horizontal component, vox, of its initial velocity?

c) What was the maximum height above its launch point?

d) As it hit the hill, what was the y-component of the velocity?

e) What is the velocity (magnitude and direction) of the projectile when the projectile hits the hill?

Explanation / Answer

Use s = s0 + v0*t - (1/2)*g*t^2

t = time after projectile is fired
g = acceleration of gravity = 9.81 m/s^2
s = altitude at time 't'
so = initial altitude = 0 (using the launch point as the origin of the coordinates)
voy = initial vertical speed

For this we have: t = 22 s ; s = 459 m
Solve for voy: voy = [s + (1/2)*g*t^2]/t = [459 + (1/2)*9.81*22^2)/22
voy = 236.68 m/s

(b) This is just the horizontal distance divided by the time of flight.
vox = 3140/22 = 142.72 m/s

(c) We need to find the time of maximum height. We can do that using voy = g*t at max height.
t = voy/g = 236.68  /9.81 = 24.13 seconds

Then use s = voy*t - (1/2)*g*t^2 = 236.68 x 24.13- 0.5 x 9.81 x (24.13)2= 2855.11 meters

(d) We know the horizontal component (since no air resistance) and that is just the same as it was at time of launch. We just need to find the vertical component. And this will be the speed it attains falling from its max height down to 459 meters.

We can use v = g*t but we first need to find the time it takes to fall.

Drop distance = 2855.11 - 459 = 2396.11 meters
s = (1/2)*g*t^2 which gives t = sqrt(2*s/g)
t = sqrt(2 x2396.11/9.81 ) = 22.10 seconds

v = g*t = 9.81*22.10 = 216.82 m/s

Speed = sqrt(216.82^2 +142.72^2) = 259.57 m/s

Tan(angle) = v/u0 = 216.82/142.72 =1.519
Angle = 56.65 degrees
This is below the horizontal so add 90 degrees to get 146.65 degrees from the upward pointing vertical

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