Projectile Motion Calculator

Last updated: 2026-04-21

Definition

Projectile motion is the trajectory of an object launched into the air and acted on only by gravity, with air resistance and other forces ignored. Galileo established the mathematical foundation in his 1638 treatise Two New Sciences: under those assumptions, the path is a parabola regardless of the projectile's mass.

This calculator takes initial velocity, launch angle, gravity, and starting height, and returns the range, peak height, time of flight, and the projectile's position and velocity at any elapsed time. The model is appropriate for introductory physics, first-order sports analysis, and trajectory prototyping in game development.

A live trajectory plot is available at Projectile Motion Simulation.

Horizontal and Vertical Decomposition

Independent Components

The central insight is that horizontal and vertical motion are independent; they share only the time variable.

Horizontally, no force acts in the vacuum model, so motion is at constant velocity:

$x(t) = v_0 \cos\theta \cdot t$

Vertically, gravity decelerates the upward motion and then accelerates the descent:

$y(t) = h_0 + v_0 \sin\theta \cdot t - \frac{1}{2} g t^2$

where $v_0$ is initial speed, $\theta$ is launch angle measured from horizontal, $g$ is gravitational acceleration, and $h_0$ is starting height above the landing surface.

The 45° Range Optimum

When launch and landing occur at the same height ($h_0 = 0$), maximum range is reached at exactly 45°:

$R = \frac{v_0^2 \sin(2\theta)}{g}$

The factor $\sin(2\theta)$ peaks at $\theta = 45°$. A direct corollary: angles equidistant from 45° produce equal ranges — 30° and 60° land at the same point, as do 20° and 70°. The symmetry is intrinsic to the sine function.

When launch height exceeds the landing surface (a cannon on a hill, a basketball release at shoulder height), the optimum angle drops below 45°; the greater the starting height, the flatter the optimal launch.

Same Trajectory, Different Gravity

The calculator includes gravity presets for the Moon (1.62 m/s²) and Mars (3.71 m/s²). Under identical launch conditions, a projectile on the Moon travels roughly six times farther than on Earth, since range is inversely proportional to $g$. Apollo 14 astronaut Alan Shepard hit two golf balls on the lunar surface in 1971, though the actual range was on the order of tens of meterstens of yards because his suited swing was constrained.

Worked Example

Take a launch at $v_0 = 20$ m/s, $\theta = 45°$, from ground level ($h_0 = 0$), under Earth gravity $g = 9.81$ m/s². The velocity components are

$v*{0x} = 20 \cos 45° \approx 14.14 \text{ m/s}, \qquad v*{0y} = 20 \sin 45° \approx 14.14 \text{ m/s}$

The time to the apex follows from the vertical velocity reaching zero, $v_{0y} - g\,t = 0$:

$t*{\text{apex}} = \frac{v*{0y}}{g} = \frac{14.14}{9.81} \approx 1.44 \text{ s}$

With equal launch and landing heights the descent mirrors the climb, so the total flight time is twice the time to the apex, about 2.88 s. The peak height and range then follow:

$h*{\max} = \frac{v*{0y}^2}{2g} = \frac{14.14^2}{2 \cdot 9.81} \approx 10.19 \text{ m}$

$R = \frac{v_0^2 \sin(2\theta)}{g} = \frac{20^2 \cdot \sin 90°}{9.81} \approx 40.77 \text{ m}$

The same range is obtained from the horizontal component carried over the full flight, $v_{0x} \cdot t_{\text{flight}} = 14.14 \cdot 2.88 \approx 40.77$ m.

Applications

Shot Put Release Angle

Elite shot putters typically release at angles between 35° and 38°, noticeably below the textbook 45°. The shot leaves the hand around 2 m above the groundaround 6.5 ft above the ground, not at ground level. With a starting height advantage, the optimum angle drops because the projectile already has additional time aloft and benefits from converting more energy into horizontal speed.

Physics Pedagogy

For introductory mechanics, varying one parameter at a time and observing the trajectory response illustrates concepts that are harder to grasp from algebraic derivations alone. Useful demonstrations include showing that 30° and 60° produce identical ranges, observing peak height grow with launch angle while range contracts past 45°, and comparing Earth–Moon–Mars trajectories side by side.

Game Development

When prototyping projectile mechanics for a game — bow and arrow, artillery, basketball — the vacuum model provides quick sanity checks for tuning. Questions such as "What initial speed is required to reach 100 m?""What initial speed is required to reach 110 yards?" can be answered before tuning the physics engine. Production games then layer on air drag, wind, and Magnus effects, but the analytic baseline is a useful reference point.

Real-World Throw Estimates

A quarterback's deep ball travels around 50–60 m at release speeds of 25–28 m/saround 55–65 yards at release speeds of 55–60 mph with launch angles near 30–35°. Substituting those values gives rough but reasonable figures — useful context when comparing the ideal calculation against actual NFL pass distances and seeing how much air resistance subtracts in practice.

Limits of the Vacuum Model

This calculator solves the vacuum model. Real projectiles experience drag, which slows them and bends the trajectory away from a perfect parabola. The effect grows with speed and surface area and shrinks with mass. A baseball thrown at 40 m/s (90 mph)at 90 mph (40 m/s) travels roughly 20–40% shorter than the vacuum prediction. Bullets, arrows, and golf balls also depart from the parabolic ideal in significant ways.

Competitive sports analysis, ballistics, and aerospace work require a model that includes drag and, for spinning projectiles, the Magnus force. The vacuum model is the appropriate starting point for understanding the geometry and serves well as a teaching tool, but high-precision predictions require more.