Projectile Motion Calculator
By Santaji GadepatilinUpdated October 27, 2026
A thrown ball, a cricket shot or a water jet all follow a curved path called a parabola. Projectile motion splits the movement into steady horizontal travel and vertical motion under gravity.
Range (horizontal distance) (m) vs Launch angle (°) — other inputs held at your current values.
How to use the projectile motion calculator
Enter the launch speed and angle to get the range, maximum height and flight time. The chart shows how range changes with angle — note the peak at 45°.
Formulas (level ground, no air resistance)
R = v² sin(2θ) ÷ g
H = v² sin²θ ÷ 2g
T = 2v sin θ ÷ g
Worked example
Launched at 20 m/s and 45°:
- Range = 400 × 1 ÷ 9.81 = 40.8 m
- Maximum height = 10.2 m
- Flight time = 2.88 s
Projectile from a height
If launched from height h₀ above the landing point, the time of flight becomes:
t = [v sin θ + √((v sin θ)² + 2 g h₀)] ÷ g
and the range is v cos θ × t. From a height, the best angle for maximum range is less than 45°.
Real-world effects
| Effect | Result |
|---|---|
| Air resistance | Shorter range, lower peak, steeper descent |
| Spin (Magnus effect) | Backspin lifts the ball; topspin makes it dip |
| Wind | Changes range and drift |
For a well-hit golf ball or a long throw, real range can be 30–50% less than the formula predicts.
Sports examples
- Long jump: athletes take off at about 18–22°, lower than 45°, because they cannot generate as much speed at steeper angles.
- Shot put: the best angle is about 37–40° because the shot is released above the ground.
- Football goal kicks: typical angles are 30–45°.
Horizontal launch
For something thrown or rolled horizontally off a table (θ = 0), time to land is √(2h ÷ g), and range is v × √(2h ÷ g). A ball rolling off a 1 m table at 2 m/s lands 0.90 m away.
Range at different angles (20 m/s)
| Angle | Range |
|---|---|
| 15° | 20.4 m |
| 30° | 35.3 m |
| 45° | 40.8 m |
| 60° | 35.3 m |
| 75° | 20.4 m |
Frequently asked questions
What angle gives the maximum range?
45° on level ground, ignoring air resistance. Complementary angles such as 30° and 60° give the same range.
Does this include air resistance?
No. Real projectiles such as balls fall shorter, and the best angle is usually a little below 45°.
Does this assume level ground?
Yes — the projectile lands at the same height it was launched from.
Why is 45° the best angle?
Range depends on sin 2θ, which is largest when 2θ = 90°. This holds only on level ground without air resistance.
Do complementary angles give the same range?
Yes. On level ground, 30° and 60° (or 20° and 70°) give the same range, but the steeper angle goes higher and takes longer.