ON A SLOPE
Range up a ramp: the launch that bisects slope and vertical, θ* = 45° + α/2, exact.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A projectile is launched from the foot of a ramp tilted at angle α, leaving at angle θ and speed v₀. How far up the slope does it land?
Two motions
Horizontal drift and vertical rise-and-fall stay independent, exactly as on flat ground. What changes is the finish line: the ground now climbs as y = x·tanα.
Solve
Set the parabola equal to the slope and solve. The positive root gives R = 2v₀²cosθ·sin(θ−α)/(g·cos²α).
Flight
The projectile arcs over and comes down onto the ramp, meeting it at the positive-time range point.
The optimum
Range is largest when the launch bisects the slope and the vertical: θ* = 45° + α/2, giving R_max = v₀²/g(1+sinα). Angles spaced equally either side of θ* travel exactly the same distance.