A projectile is anything that flies freely after launch — a kicked ball, a stream of water, a stone thrown off a ledge — with gravity as the only force acting on it. This calculator traces the whole flight from the launch speed, launch angle and launch height, then reports the range, peak height, flight time and impact speed. It can also work backward: tell it how far away the target is and it finds the launch speed or angle that lands there.
How to use the projectile motion calculator
- Under What do you want to find? choose a mode:
- Trajectory from launch speed and angle — the standard forward problem.
- Launch speed needed to land at a distance — you fix the angle and the target distance.
- Launch angle needed to land at a distance — you fix the speed and the target distance.
- Fill in the visible fields: Launch speed (v₀), Launch angle (θ) measured above the horizontal, and Target distance (R).
- Enter the Launch height (h₀) above the landing level, or leave it at 0 for level ground.
- Pick the Gravity: Earth (9.80665 m/s²), Moon, Mars, Jupiter, or a custom value.
- Choose the units for distances and speeds. The result shows the range, time of flight, maximum height and when it is reached, both velocity components and the impact speed and angle. Below that you get a scaled trajectory diagram and a table of position and vertical velocity at eleven evenly spaced moments.
Projectile motion formulas
The trick is to split the launch velocity into two independent motions: steady drift sideways and free fall up and down.
x(t) = vxt · y(t) = h0 + vy0t − ½gt²
Setting y = 0 gives the time of flight, and the rest follows:
R = vxT · H = h0 + vy0² ÷ 2g
On level ground (h0 = 0) these shrink to T = 2v0 sin θ ÷ g and R = v0² sin 2θ ÷ g.
Worked example: 20 m/s at 45°
A ball leaves the ground at 20 m/s, 45° above the horizontal, on Earth.
vx = vy0 = 20 × 0.70711 = 14.142 m/s
T = 2 × 14.142 ÷ 9.80665 = 2.884 s
R = 14.142 × 2.884 = 40.79 m
H = 14.142² ÷ (2 × 9.80665) = 10.20 m, reached at 1.442 s
The ball lands at 20 m/s, 45° below the horizontal — a mirror image of the launch, as it always is on level ground. Switch the gravity to the Moon and the same throw flies 246.9 m and stays up 17.46 s.
Solving for the launch speed or angle
In speed mode the calculator uses v0 = √(gR² ÷ [2cos²θ (h0 + R tan θ)]). To land 30 m away at 45° on level ground you need 17.15 m/s.
In angle mode the landing condition becomes a quadratic in tan θ, so there are usually two answers. At 20 m/s with a 30 m target, a flat shot at 23.67° arrives in 1.638 s after peaking at 3.29 m, while a lofted shot at 66.33° takes 3.736 s and climbs to 17.11 m. The diagram draws the lofted path as a dashed line. When the target sits exactly at the maximum range the two answers merge into one, and beyond it the calculator reports the farthest reachable distance instead.
Why 45° is only the level-ground answer
The 45° rule comes from maximizing sin 2θ, which assumes the projectile lands at the height it started. Launch from above the landing level and the extra fall time favors a flatter shot. At 20 m/s from a 10 m platform:
| Angle | Range |
|---|---|
| 35° | 49.41 m |
| 40° | 49.78 m |
| 45° | 49.10 m |
The best angle here is about 39.3°, giving 49.79 m. The higher the platform relative to the launch speed, the lower the best angle.
The model ignores air resistance and assumes constant gravity over a flat landing level. That suits slow, dense objects over short distances; for fast or light objects such as table-tennis balls or badminton shuttles, real ranges are much shorter.
For straight-down drops use the free fall calculator, and for one-dimensional motion with any three known values try the SUVAT calculator.
Frequently asked questions
What launch angle gives the maximum range?
On level ground with no air resistance, 45°. From a raised launch point the best angle is lower: at 20 m/s from 10 m up, 45° reaches 49.10 m while about 39.3° reaches 49.79 m. Air drag lowers the best angle further for real balls.
Why are there two launch angles that hit the same spot?
For any target inside the maximum range, a flat shot and a lofted shot both land there. On level ground they are complementary: at 20 m/s, 30° and 60° both travel 35.32 m, but the 60° shot stays up 3.53 s instead of 2.04 s.
Does the projectile motion calculator include air resistance?
No. It assumes a vacuum, constant gravity and a flat landing level, so the horizontal velocity never changes. Real balls, arrows and shells fall short of these figures, and the gap grows with speed.
What does 'out of reach' mean in the angle mode?
The target is farther than the projectile can go at that launch speed, whatever the angle. The limit is R = (v₀/g)√(v₀² + 2gh₀), which for 20 m/s on level Earth ground is 40.79 m.
Can I launch downward, for example from a cliff?
Yes. Enter a launch height above zero and a negative angle; angles must be between −90° and 90°. From ground level, an angle at or below the horizontal is rejected because the projectile would land immediately.