Projectile Motion Calculator

Launch speed, angle and height in; range, maximum height, time of flight and impact velocity out. No air resistance, any gravity.

Earth 9.81 · Moon 1.62 · Mars 3.71
R
Horizontal range
40.77m
H
Maximum height
10.19m
reached at t = 1.44 s
T
Time of flight
2.883s
v
Impact speed
20m/s
45° below horizontal
vₓ
Horizontal velocity (constant)
14.14m/s
v_y
Initial vertical velocity
14.14m/s

Why 45° goes furthest, and why 30° ties with 60°

The solid arc is your angle; the dashed arc is its complement (90° − θ). Drag the angle and watch them land on the same spot.

35.3 m

θ = 30°: range 35.3 m, height 5.1 m. θ = 60°: range 35.3 m, height 15.3 m. Same range, because sin 2θ = sin(180° − 2θ).

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0/1500

Worked example

A ball is kicked at 20 m/s at 45° on level ground. vₓ = v_y = 14.14 m/s. Time of flight T = 2 × 14.14 ÷ 9.81 = 2.88 s. Range R = 14.14 × 2.88 = 40.8 m. Maximum height H = 14.14² ÷ (2 × 9.81) = 10.2 m.

The key idea: horizontal and vertical motion are independent. Horizontally nothing pushes the ball, so vₓ never changes. Vertically it is a free-fall problem with an upward start.

Frequently asked questions

What are the projectile motion formulas?
Split the launch velocity: vₓ = v cos θ, v_y = v sin θ. Horizontally x = vₓt (no acceleration). Vertically y = h + v_y t − ½gt². From these, level-ground range R = v² sin 2θ ÷ g, max height H = v_y² ÷ 2g and flight time T = 2v_y ÷ g.
What angle gives the maximum range?
45° on level ground, because sin 2θ is largest at 2θ = 90°. Launched from a height, the best angle is a bit less than 45°. With air resistance it is lower still, around 35–40° for a ball.
Why do 30° and 60° give the same range?
Range depends on sin 2θ, and sin 60° = sin 120°. Any two complementary angles (adding to 90°) land at the same spot on level ground; the steeper one flies higher and stays up longer.
Does the mass of the projectile matter?
Not without air resistance: all objects fall with the same acceleration g. That is the main simplification here; real balls, especially light or fast ones, fall short of these numbers.
How do I handle a launch from a cliff?
Enter the launch height. The flight time then comes from solving h + v_y t − ½gt² = 0 for the positive root, which this calculator does.