About the Projectile Motion Calculator
This projectile motion calculator helps you analyze the parabolic trajectory of any launched object. Enter the initial velocity, launch angle, and optional launch height to instantly find the range, time of flight, maximum height, and velocity components — in both metric and imperial units.
What is Projectile Motion?
Projectile motion is the motion of an object launched into the air that travels under the influence of gravity alone (air resistance is neglected). Any object — a ball, an arrow, a cannonball — that is given an initial velocity and then follows a curved path due to gravity is in projectile motion.
The path traced by such an object is a parabola. Projectile motion can always be split into two independent components:
Horizontal Motion
Constant velocity — no acceleration (gravity acts only vertically). The object travels the same horizontal distance each second.
Vertical Motion
Uniformly accelerated by gravity (g = 9.81 m/s² or 32.174 ft/s²). The object slows as it rises, stops momentarily at peak height, then accelerates downward.
Projectile Motion Equations
Given initial speed V₀, launch angle α, and initial height h:
Vx = V₀ · cos(α) — horizontal velocity component (constant)
Vy₀ = V₀ · sin(α) — initial vertical velocity component
t = (Vy₀ + √(Vy₀² + 2gh)) / g — time of flight
R = Vx · t — horizontal range
Hmax = h + Vy₀² / (2g) — maximum height
Step-by-step Example
A ball is kicked at 20 m/s at a 45° angle from ground level (h = 0).
Vx = 20 · cos(45°) = 14.14 m/s
Vy₀ = 20 · sin(45°) = 14.14 m/s
t = 2 · Vy₀ / g = 2 · 14.14 / 9.81 = 2.88 s
R = 14.14 · 2.88 = 40.74 m
Hmax = 14.14² / (2 · 9.81) = 10.19 m
The Optimal Angle for Maximum Range
When launching from flat ground (h = 0), the angle that produces the greatest horizontal range is 45°. At this angle, horizontal and vertical velocity components are equal, maximising both the time in the air and the horizontal speed simultaneously.
For launches from an elevated position (h > 0), the optimal angle for maximum range is less than 45° — the higher the launch point, the shallower the ideal angle.
Time of Flight Formula
The time of flight depends on whether the projectile is launched from ground level or an elevated position:
From Ground (h = 0)
t = 2 · Vy₀ / g
The flight is perfectly symmetrical — ascent and descent take equal time.
From Elevation (h > 0)
t = (Vy₀ + √(Vy₀² + 2gh)) / g
The extra height adds to flight time because the projectile must travel further to reach ground.
Metric vs Imperial Units
| Quantity | Metric (SI) | Imperial (US) |
|---|---|---|
| Speed / Velocity | m/s (metres per second) | ft/s (feet per second) |
| Distance / Height | m (metres) | ft (feet) |
| Gravity (g) | 9.81 m/s² | 32.174 ft/s² |
| Time | seconds (s) | seconds (s) |
Real-world Applications
Sports
- Football / Soccer — optimising kick angle for maximum distance
- Basketball — calculating the arc of a shot
- Golf — club angle vs. ball flight distance
- Javelin / Shot put — finding optimal release angle
Engineering & Science
- Ballistics — predicting artillery shell trajectories
- Rocketry — early-stage flight analysis
- Water fountains — nozzle angle design
- Stunt design — calculating safe landing zones
Frequently Asked Questions
- What angle gives the maximum range? — 45° for a flat launch surface. For elevated launches, a slightly smaller angle maximises range.
- Does mass affect projectile motion? — No. All objects fall at the same rate regardless of mass (assuming no air resistance), so mass does not appear in the equations.
- What is the gravity value used? — 9.81 m/s² for metric, 32.174 ft/s² for imperial. These are standard Earth surface values.
- Can I use this for horizontal projectile motion (angle = 0)? — Yes, but only if the initial height is greater than zero. A horizontal launch from the ground is not physical (the object is already on the ground).
- What happens at 90°? — The projectile goes straight up with no horizontal range. Time of flight is 2 · V₀ / g and maximum height is V₀² / (2g).
- Is air resistance included? — No. This calculator uses the standard physics model that neglects air resistance, giving idealised results. Real trajectories will be shorter due to drag.