Projectile motion calculator
Enter launch speed, angle and height, get range and flight time.
How far does a projectile travel?
On level ground, range = v² × sin(2θ) ÷ g. Launched at 20 m/s and 45°, it travels 400 ÷ 9.80665 = 40.8 meters, ignoring air resistance.
Which angle goes farthest?
45° on level ground. Launching from a height, the best angle is a little lower.
How long is it in the air?
From the ground, t = 2 × v × sin θ ÷ g: 2.88 seconds for 20 m/s at 45°.
How high does it go?
h = (v × sin θ)² ÷ 2g: 10.2 m for the same throw.
Why is the real distance shorter?
Air resistance slows the projectile — a lot for a light ball, less for a heavy one. These results are the ideal, airless case.
Why do two different angles give the same range?
Two launch angles that add up to 90° land at the same spot when the ground is level (OpenStax University Physics, Volume 1, section 4.3). At 20 m/s, 30° and 60° both reach 35.3 m, but the 30° shot peaks at 5.1 m and the 60° shot at 15.3 m, and the steeper one stays in the air longer. OpenStax Example 4.9: a 90 m golf shot needs 31.9 m/s at 30° and 37.0 m/s at 70°.
What if the target is higher or lower than the launch point?
The calculator measures from a launch height above the landing level, so it covers a throw from a roof or a cliff. If the object must land higher than it started, the simple range and flight-time formulas no longer apply, because they hold only for launch and impact on the same horizontal surface (OpenStax, section 4.3); you solve a quadratic for the time instead. OpenStax Example 4.8 hits a tennis ball at 30 m/s and 45° into stands 10.0 m higher and takes the longer root, 3.79 s.
How fast is the projectile moving when it lands?
The horizontal speed never changes; the vertical speed does. For a throw from height h the landing speed is √(vₓ² + v_y² + 2gh). In the example below that is 24.99 m/s (82.0 ft/s), slightly more than the 80 ft/s launch speed because the object lands 5 ft lower than it started.
The formulas and what they assume
OpenStax University Physics, Volume 1, section 4.3, treats the horizontal and vertical motions separately. Horizontally the speed is constant, vₓ = v cos θ. Vertically the acceleration is −g and the starting speed is v sin θ. From level ground the time of flight is T = 2v sin θ ÷ g (Eq. 4.24), the maximum height is (v sin θ)² ÷ 2g and the range is R = v² sin 2θ ÷ g (Eq. 4.26). With a launch height the calculator solves the vertical motion for the flight time and multiplies it by vₓ. All of this neglects air resistance and takes g as constant (9.80665 m/s²).
Worked example: 80 ft/s at 40°, released 5 ft above the ground
In SI the speed is 80 × 0.3048 = 24.38 m/s and the height 1.524 m. The components are vₓ = 18.68 m/s and v_y = 15.67 m/s. The object reaches its top after 1.60 s, 46.1 ft above the ground, and lands after 3.29 s, 201.7 ft away.
Common mistakes
The most common is believing 45° is always best. It is best only for level ground and no air; with a launch height the best angle is a little lower, and with air resistance OpenStax notes it is also smaller. Another is mixing the speed unit: 80 read as m/s instead of ft/s is a factor of 3.28 too fast and the range is nearly eleven times too long (3.28² = 10.8). Third, the speed you enter is the speed along the launch direction, not its horizontal part. And the angle is in degrees: a calculator left in radian mode reads 40 as 40 radians, about 2,292°.
Sources: OpenStax University Physics, Volume 1, section 4.3 (projectile motion, Eq. 4.24–4.26; Examples 4.8 and 4.9); standard gravity 9.80665 m/s² is exact (NIST SP 811, Appendix B.8). Related: free fall calculator, kinematics calculator, terminal velocity calculator and speed calculator.