initial vertical speed = vy = v sin theta
initial horizontal speed = vx = v cos theta
range = flight time * horizontal speed
r = t * vx
Solve for the flight time
t = r / vx = r / (v cos theta)
The height as a function of time is:
h(t) = vy t - 1/2 gt^2
Plugging in our flight time:
h = v sin (theta) r / (v cos theta) - 1/2 g (r / (v cos theta))^2
= r tan (theta) - g r^2 / (2 v^2 cos(theta)^2)
b) Use conservation of energy
initial KE = 1/2 mv^2
=
final KE + PE = 1/2 m vf^2 + mgh
Solve for your landing speed:
vf = sqrt (v^2 - 2gh)
= sqrt (v^2 - 2 g (r tan (theta) - g r^2 / (2 v^2 cos(theta)^2))
Have fun plugging in numbers.