physics trailbike problem- help??

Question

A ditch 2.29 m wide crosses a trailbike path. An upward incline of 15.5 degrees has been built up on the approach so that the top of the incline is level with the top of the ditch. What is the minimum speed (in m/s) at which a trailbike must be moving to completely clear the ditch? (Add 1.52 m to the range for the back of the bike to clear the ditch safely.)

Answer 1

Call the initial speed v_i, and note that the initial velocity has components:
v_ix = v_i*cos(15.5)
v_iy = v_i*sin(15.5)

Since launch and landing are at the same elevation, the initial and final speeds will be identical, and the velocity v_f will be given by components:
v_fx = v_ix (no horizontal acceleration)
v_fy = -v_iy (bike is falling now, not rising)

The time of flight t is the time it takes gravity (acceleration -g) to change vertical velocity from v_iy to v_fy:
t = (v_fy - v_iy) / -g = 2*v_iy / g

But t is also the time it takes to cover a distance x (sum of ditch plus safety distance) at a constant speed v_ix:
x = v_ix * t

The last two equations have only two unknowns, v_ix and v_iy. Solve and compute v_i = sqrt(v_ix^2 + v_iy^2)

This is a lot messier in ASCII than it is on a drawing. Always make a drawing.