A gunman standing on a sloping ground fires up the slope. The initial speed of the bullet is v0= 360 m/s. The slope has an angle α= 28° from the horizontal, and the gun points at an angle θ from the horizontal. The gravitational acceleration is g=10 m/s2. Let θ = 59°. Find the maximum range, lmax, of the bullet along the slope.

Question

A gunman standing on a sloping ground fires up the slope. The initial speed of the bullet is v0= 360 m/s. The slope has an angle α= 28° from the horizontal, and the gun points at an angle θ from the horizontal. The gravitational acceleration is g=10 m/s2.  Let θ = 59°.  Find the maximum range, lmax, of the bullet along the slope.

Steve · Answer

As you recall, the height of the bullet is given by

y(x) = tanθ x - g/(2(v cosθ)^2) x^2
and of course, the height of the slope is
y(x) = tanα x

So, using your numbers, you just need to find where the two curves intersect:

tan59° x - 5/(360 cos59°)^2 x^2 = tan28° x
I get x = 7787.2
Of course, that is the horizontal distance; I'm sure you can use that to find the distance up the slope ...