# Projectile Motion Calculator

Calculate projectile range, maximum height, and time of flight. Enter initial velocity, launch angle, and gravity. Uses standard kinematic equations.

## What this calculates

Projectile motion describes the curved path of an object launched into the air under the influence of gravity alone (ignoring air resistance). This calculator uses the standard kinematic equations to compute the range, maximum height, time of flight, and velocity components for any launch speed, angle, and gravitational field.

## Inputs

- **Initial Velocity** (m/s) — min 0
- **Launch Angle** (°) — min 0, max 90 — Angle above the horizontal (0° to 90°)
- **Gravitational Acceleration** (m/s²) — min 0 — Earth = 9.81, Moon = 1.62, Mars = 3.72

## Outputs

- **Range** (m) — Horizontal distance traveled
- **Maximum Height** (m) — Peak height above launch point
- **Time of Flight** (s) — Total time in the air
- **Time to Apex** (s) — Time to reach maximum height
- **Horizontal Velocity** (m/s) — Constant horizontal velocity component
- **Initial Vertical Velocity** (m/s) — Initial vertical velocity component

## Details

The three key equations for projectile motion launched from ground level are:

  - Range: R = v²sin(2θ) / g

  - Maximum Height: H = v²sin²(θ) / (2g)

  - Time of Flight: T = 2v·sin(θ) / g

Where v is the initial velocity, θ is the launch angle above the horizontal, and g is the acceleration due to gravity (9.81 m/s² on Earth).

Key insights:

  - Maximum range occurs at 45° (when sin(2θ) = 1)

  - Complementary angles (e.g., 30° and 60°) give the same range but different heights and flight times

  - The horizontal velocity remains constant throughout flight (no air resistance)

  - At the apex, vertical velocity is zero but horizontal velocity is unchanged

You can also explore how projectiles behave on different celestial bodies by adjusting the gravity parameter. The Moon's weaker gravity (1.62 m/s²) makes projectiles fly about 6 times farther than on Earth for the same launch conditions.

## Frequently Asked Questions

**Q: What angle gives maximum range?**

A: A launch angle of 45° gives the maximum range for a given initial velocity (when launching and landing at the same height). This is because sin(2×45°) = sin(90°) = 1, maximizing the range formula.

**Q: Does air resistance affect projectile motion?**

A: Yes, significantly. Air resistance (drag) reduces both range and maximum height, and the optimal angle drops below 45°. This calculator assumes ideal conditions with no air resistance.

**Q: What happens to a projectile at its highest point?**

A: At the apex of its trajectory, the vertical velocity is momentarily zero while the horizontal velocity remains unchanged. The object still has kinetic energy from horizontal motion and maximum gravitational potential energy.

**Q: Why do complementary angles give the same range?**

A: Because range depends on sin(2θ), and sin(2×30°) = sin(60°) is the same as sin(2×60°) = sin(120°). The two angles trade off height for flight time but produce the same horizontal distance.

**Q: How would this calculator work on the Moon or Mars?**

A: Change the gravity value. The Moon has g = 1.62 m/s² and Mars has g = 3.72 m/s². Lower gravity means greater range, higher maximum height, and longer flight time for the same launch conditions.

---

Source: https://vastcalc.com/calculators/physics/projectile-motion
Category: Physics
Last updated: 2026-04-21