# Projectile Motion

> Launch angle, speed and gravity fully determine a ballistic arc: x moves at constant speed while y accelerates downward.

- Category: Physics › Mechanics
- Level: core
- Tags: kinematics, gravity, ballistics, trajectory
- Updated: 2026-11-01
- URL: https://gdcs.coody.app/physics/mechanics/projectile-motion

## What it is

A projectile is any body moving under gravity alone. Split its velocity into components: the horizontal part `v0·cos θ` never changes (no drag), while the vertical part `v0·sin θ` loses `g` metres per second every second.

Because the two axes are independent, the path is a parabola. Flight time, peak height and range all follow from the launch speed, angle and gravity, so you can predict a landing point before the shot happens.

In games this is how grenades, arrows, jump arcs and aim-assist trajectory previews are computed. It is also the reference you check a [fixed timestep](https://gdcs.coody.app/systems/simulation/fixed-timestep) integrator against.

## Key formulas

- **Horizontal position:** $x(t) = v_0 \cos\theta \, t$
- **Vertical position:** $y(t) = v_0 \sin\theta \, t - \tfrac{1}{2} g t^2$
- **Flight time (flat ground):** $T = \frac{2 v_0 \sin\theta}{g}$
- **Range:** $R = \frac{v_0^2 \sin 2\theta}{g}$ — Maximum at 45° on flat ground, with no drag.
- **Peak height:** $H = \frac{(v_0 \sin\theta)^2}{2g}$

## Rules of thumb

- Complementary angles (30° and 60°) land at the same spot; the steeper one stays in the air longer.
- Earth `g = 9.81 m/s²`. Many platformers use 2–3× that for snappier jumps.
- To hit a target, solve the range equation for θ. Two solutions mean a low and a high lob.

## Implementation

Sample an arc for a trajectory preview (Three.js):

```ts
import * as THREE from "three"

export function trajectory(v0: number, angle: number, g = 9.81, samples = 64) {
  const vx = v0 * Math.cos(angle)
  const vy = v0 * Math.sin(angle)
  const T = (2 * vy) / g
  const points: THREE.Vector3[] = []
  for (let i = 0; i <= samples; i++) {
    const t = (i / samples) * T
    points.push(new THREE.Vector3(vx * t, vy * t - 0.5 * g * t * t, 0))
  }
  return new THREE.BufferGeometry().setFromPoints(points) // draw with THREE.Line
}
```

## Common mistakes

- Mixing degrees and radians: `Math.sin(45)` is not sin 45°.
- Integrating with explicit Euler at a variable frame rate, which makes the landing point depend on FPS.
- Forgetting launch height: the flat-ground formulas assume the projectile lands at the height it started from.
- Adding drag later and still trusting the analytic range. Once there is drag, simulate numerically.

## Interactive visual

The orange ball follows the cyan arc and the yellow line is its instantaneous velocity. Change the angle, launch speed and gravity to reshape the arc; the readout shows flight time and range.

## Prompt

```text
Implement a projectile trajectory preview in TypeScript with Three.js. Given launch speed v0 (m/s), angle θ (radians) and gravity g, compute x(t)=v0·cosθ·t and y(t)=v0·sinθ·t−½gt², sample 64 points until it returns to launch height (T=2v0·sinθ/g), and draw them with THREE.Line. Also return range R=v0²·sin2θ/g and peak height H=(v0·sinθ)²/(2g). Keep angles in radians, make the sampler a pure function, and add a unit test checking that 30° and 60° give the same range.
```

## Related

- [Fixed Timestep](https://gdcs.coody.app/systems/simulation/fixed-timestep.md) — Run physics in constant dt slices from an accumulator, clamp long frames, and interpolate between the last two states for smooth, repeatable motion.
- [Projectile Data Lab](https://gdcs.coody.app/physics/mechanics/projectile-data-lab.md) — Repeat a launch with a little speed error and the landings form a distribution. Mean, spread and why R ∝ v² shifts the average: the maths of weapon spread.
- [Kepler's Laws](https://gdcs.coody.app/physics/mechanics/keplers-laws.md) — Orbits are ellipses with the star at one focus, sweep equal areas in equal times, and have T² ∝ a³. Place planets on rails exactly with Kepler's equation.

## Further reading

- [OpenStax University Physics — Projectile Motion](https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion)

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Source: Game Development Cheat Sheet (https://gdcs.coody.app)