Projectile Motion Calculator
Range, maximum height, time of flight and trajectory plot — with step-by-step working.
Formula
Vertical: y(t) = h + v₀·sin(θ)·t − ½·g·t²
Time of flight (level ground, h = 0): T = 2·v₀·sin(θ) / g
Maximum height: H = h + (v₀·sin θ)² / (2g)
Range (level ground): R = v₀²·sin(2θ) / g
How to use
- Enter the initial launch speed in m/s, km/h, mph or ft/s.
- Enter the launch angle above the horizontal (degrees or radians).
- Enter the launch height above the landing surface. Use 0 for level-ground launches.
- Pick the gravity for your scenario — Earth, Moon, Mars or a custom value.
- Press Calculate. The trajectory is drawn and every output is given with units.
Physics behind projectile motion
A projectile, once released, is pulled only by gravity. The classical idealisation (no air resistance) splits the motion into two independent axes: horizontal, where velocity is constant, and vertical, where acceleration equals −g. This decomposition is the cornerstone of kinematics and is the reason the path is a parabola.
At launch, the velocity vector has magnitude v₀ and makes an angle θ with the horizontal. The components are vx = v₀·cos(θ) and vy = v₀·sin(θ). Because no horizontal force acts, vx stays constant throughout the flight; the projectile travels a horizontal distance x = vx·t. Vertically, gravity reduces vy linearly; at the top of the flight vy = 0. The height grows then shrinks, producing the familiar symmetric parabola for level-ground launches.
Range is maximised at 45° on level ground because sin(2θ) reaches its peak at θ = 45°. When the launch point is above the landing surface (h > 0), the optimal angle drops below 45° — the extra fall time gives the horizontal component more time to work.
Worked example
v₀ = 20 m/s, θ = 45°, h = 0, g = 9.80665 m/s²
vx = 20 cos 45° = 14.142 m/s vy = 20 sin 45° = 14.142 m/s T = 2·14.142 / 9.80665 ≈ 2.884 s H = 14.142² / (2·9.80665) ≈ 10.19 m R = 20²·sin(90°) / 9.80665 ≈ 40.77 m
Common mistakes
- Mixing degrees and radians. The formulas use the sine of the angle — make sure your calculator (and this tool's unit selector) agrees.
- Using g = 10 as a rounded value. For exam working 9.81 or 9.80665 gives more accurate answers; our tool defaults to CODATA 9.80665.
- Forgetting that R = v²·sin(2θ)/g is only valid when the launch height equals the landing height.
- Negative heights. The tool allows h = 0 or positive values; the projectile lands at y = 0.
Related tools
FAQs
What is projectile motion?
Projectile motion is the curved path followed by an object launched with some initial velocity and affected only by gravity. Horizontal motion is uniform; vertical motion is uniformly accelerated downward by g.
What launch angle gives the maximum range?
On level ground, 45° — because sin(2θ) peaks at θ = 45°. When launching from a higher point, the optimum angle is slightly below 45°.
Does mass affect the range?
No. Without air resistance, the trajectory depends only on initial velocity, angle, height and gravity.
Does this calculator include air resistance?
No. Use the Projectile Launcher with Drag simulator for trajectories that include aerodynamic drag.
How is the time of flight calculated with a non-zero launch height?
By solving the quadratic h + v·sinθ·t − ½gt² = 0 for t and keeping the positive root.
Which gravity values can I use?
Earth (9.80665), Moon (1.62), Mars (3.71) or any custom value.