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Projectile Motion on an Incline Calculator

Find time of flight, range, and horizontal distance for an object launched at an angle onto an upward or downward sloping incline, generalizing flat-ground projectile motion.

Time of flight
2.1107 s
Range along the incline
30.9021 m
Horizontal distance
29.8492 m

How it works

This site's existing Projectile Motion calculator only lands back on flat ground. Here, the object launches at speed v and angle θ above horizontal, but lands on a slope tilted at angle α above horizontal (positive for uphill, negative for downhill). Substituting the usual trajectory x = v·cos(θ)·t, y = v·sin(θ)·t − ½gt² into the slope's own line y = x·tan(α) and solving for the nonzero time gives time of flight t = 2v·sin(θ−α) ÷ (g·cos(α)), and the slant distance traveled along the incline R = 2v²·cos(θ)·sin(θ−α) ÷ (g·cos²(α)). The horizontal distance covered is just v·cos(θ)·t, the unchanged horizontal velocity times that time of flight.

Setting α = 0 recovers the flat-ground formulas exactly (t = 2v·sin(θ)/g and R = v²·sin(2θ)/g) — this calculator's own test suite checks that reduction directly against the site's flat-ground Projectile Motion calculator. A launch angle at or below the incline angle has no forward solution (the shot would have to travel into the slope rather than away from it), so that combination is rejected rather than returning a nonsensical result.

FAQ

Why is the range longer for a downhill shot than the same shot on flat ground?

The projectile has farther to fall before it reaches the (lower) ground, so gravity gets more time to act and the horizontal distance covered keeps growing for longer. The formula captures this directly: with α negative, cos(α) is still positive but sin(θ−α) grows larger than sin(θ) alone, increasing both the time of flight and the range compared to the flat-ground case.

Does the launch angle for maximum range still default to 45°?

No — 45° only maximizes range on flat ground. Once the landing point is tilted, the optimal launch angle shifts toward the slope: it increases above 45° for an uphill shot and decreases below 45° for a downhill one, which is why this calculator leaves the launch angle fully adjustable rather than assuming 45° is still optimal.

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