Projectile Motion With Height Difference (Velocity and Angle)

From launch speed, launch angle, and the height difference between launch and landing, compute horizontal range, time of flight, peak height, and landing speed.

Input

Enter the launch speed, launch angle, and the height difference between launch and landing to compute the horizontal range, time of flight, and more.

Launch speed v0

m/s

Speed at the instant of launch (greater than 0).

Launch angle θ

deg

Angle from the horizontal, between 0 and 90 degrees.

Height difference Δh

Height difference between launch and landing. Positive when the landing point is lower, negative when it is higher.

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

Result

Horizontal range

49.784689m

Time of flight

3.249465 s

Peak height (from launch)

8.426444 m

Time to peak

1.310922 s

Landing speed

24.415835 m/s

This is an ideal model that ignores air resistance. The time of flight is the positive root of (1/2) g t² − v0 sinθ·t − Δh = 0, and the horizontal range is v0 cosθ times the time of flight.

How it works

The launch point is taken as the origin and air resistance is ignored in this ideal projectile model. The gravitational acceleration is the standard value g = 9.80665 m/s².
The vertical position relative to the launch point is y(t) = v0 sinθ·t − (1/2) g t², and landing occurs when y = −Δh, where Δh is positive when the landing point is lower.
Solving this for t gives (1/2) g t² − v0 sinθ·t − Δh = 0, whose positive root is the time of flight T.
The horizontal range is R = v0 cosθ·T, the peak height above the launch point is H = (v0 sinθ)² ÷ (2g), and the time to the peak is v0 sinθ ÷ g.
The landing speed is obtained by combining the horizontal component v0 cosθ with the vertical component at landing. Real distances are shorter due to air resistance and wind.

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