Projectile Motion with Height Difference (from Altitude and Distance)

Enter the peak height, horizontal range, and height difference to back out the launch speed, launch angle, and flight time, with a sketch of the trajectory.

Input

Taking the launch point as the origin, the launch speed and angle are derived from the peak height, horizontal range, and height difference (no air resistance).

Peak height h

Height of the apex measured from the launch point (greater than 0)

Horizontal range R

Horizontal distance to the landing point (greater than 0)

Height difference

How much lower the landing point is than the launch point. Positive means downhill, negative means uphill.

The gravitational acceleration is the standard value g = 9.80665 m/s².

Result

Launch speed v0

24.269489m/s

Launch angle θ

54.693724deg

Flight time

4.277624 s

Horizontal speed vx

14.026479 m/s

Vertical speed vy

19.805706 m/s

Computed from vy = sqrt(2 g h), tUp = vy / g, tDown = sqrt(2 (h + difference) / g), T = tUp + tDown, vx = R / T, v0 = sqrt(vx² + vy²), theta = atan2(vy, vx).

How it works

The launch point is taken as the origin and air resistance is ignored. The gravitational acceleration is the standard value g = 9.80665 m/s².
The vertical launch speed follows from the peak height h as vy = sqrt(2 g h). The rise time from launch to apex is tUp = vy / g.
The landing point sits a height difference below the launch point. The drop from the apex is h plus that difference, and the descent time is tDown = sqrt(2 (h + difference) / g).
The flight time T is the sum of the rise and descent times, T = tUp + tDown. The horizontal speed is vx = R / T, the launch speed is v0 = sqrt(vx² + vy²), and the launch angle is theta = atan2(vy, vx).
A positive height difference means the landing point is lower than the launch point, and a negative one means it is higher. Inputs where the peak height plus the difference is zero or below cannot be computed.

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