# Projectile motion: how far and high?

Works out a projectile’s range, time of flight, maximum height and impact speed from its launch speed, angle and height, with no air resistance, and draws its path.

- Page: https://www.acalculator.org/physics/projectile-motion-calculator
- JSON spec: https://www.acalculator.org/physics/projectile-motion-calculator.json
- Version: 2b10e270676a

## Default answer

Example with the default inputs (Launch speed 65.6167979002625 ft/s, Launch angle 45 °, Launch height 0 ft, Gravity (m/s²) 9.80665): It lands 133.821 ft away after 2.88419 s, reaching 33.4553 ft at the top.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| v | Launch speed | How fast the projectile leaves the ground or the launcher. |
| a | Launch angle | The angle above the horizontal: 0° is flat, 90° is straight up. |
| h | Launch height | How high above the landing level it is launched (0 for level ground). |
| g | Gravity (m/s²) | The acceleration of gravity in m/s²: 9.80665 on Earth, 9.8 in many textbooks, 1.62 on the Moon. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| range | Range | The horizontal distance from the launch point to where it lands. |
| time | Time of flight (s) | How long it is in the air, in seconds. |
| peak | Maximum height | The highest point above the landing level: h₀ + (v₀ sin θ)² ÷ 2g. |
| apexTime | Time to the top (s) | v₀ sin θ ÷ g, in seconds. |
| impact | Impact speed | The speed when it lands: √(v₀² + 2 g h₀). |
| impactAngle | Impact angle (° below horizontal) | How steeply it comes down when it lands, in degrees below the horizontal. |

## Method

vₓ = v₀ cos θ, vᵧ = v₀ sin θ; t = (vᵧ + √(vᵧ² + 2gh₀)) ÷ g; R = vₓ t; h_max = h₀ + vᵧ² ÷ 2g; v_impact = √(v₀² + 2gh₀).

## Assumptions

- No air resistance, a flat landing level, and the same gravity everywhere along the path.
- The launch height is above the landing level; the launch angle is 0° to 90° above the horizontal.

## Worked examples

1. v = 70, a = 1.308997, h = 0, g = 9.8 gives peak = 233.253175, apexTime = 6.89947, range = 250. Source: OpenStax, University Physics Volume 1, §4.3 Projectile Motion (x = v₀ₓt, y = v₀ᵧt − ½gt², h = v₀ᵧ² ÷ 2g, R = v₀² sin 2θ₀ ÷ g), https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion (Example 4.7: h = 233 m, t = 6.90 s to the top).
2. v = 20, a = 0.785398, h = 0, g = 9.80665 gives range = 40.788649, time = 2.884193, peak = 10.197162. Source: OpenStax, University Physics Volume 1, §4.3 Projectile Motion (x = v₀ₓt, y = v₀ᵧt − ½gt², h = v₀ᵧ² ÷ 2g, R = v₀² sin 2θ₀ ÷ g), https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion.
3. v = 10, a = 0, h = 20, g = 9.8 gives time = 2.020305, range = 20.203051, impact = 22.181073. Source: OpenStax, University Physics Volume 1, §4.3 Projectile Motion (x = v₀ₓt, y = v₀ᵧt − ½gt², h = v₀ᵧ² ÷ 2g, R = v₀² sin 2θ₀ ÷ g), https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion.
4. v = 30, a = 1.570796, h = 0, g = 9.80665 gives range = 0, time = 6.118297, impactAngle = 90. Source: OpenStax, University Physics Volume 1, §4.3 Projectile Motion (x = v₀ₓt, y = v₀ᵧt − ½gt², h = v₀ᵧ² ÷ 2g, R = v₀² sin 2θ₀ ÷ g), https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion.

## FAQ

### What is the range formula for projectile motion?

On level ground, R = v₀² × sin 2θ ÷ g. A projectile at 20 m/s and 45° on Earth lands 20² × sin 90° ÷ 9.80665 = 40.79 m away. From a height the page uses the full flight time instead, because the level-ground formula no longer holds.

### What angle gives the longest range?

45° on level ground, with no air resistance, because sin 2θ is largest at θ = 45°. From a raised launch the best angle is a little less than 45°. Angles that add to 90°, such as 30° and 60°, give the same level-ground range.

### How long is a projectile in the air?

t = (v₀ sin θ + √((v₀ sin θ)² + 2 g h₀)) ÷ g, where h₀ is the launch height. On level ground this is 2 v₀ sin θ ÷ g.

### How high does a projectile go?

The top of the path is (v₀ sin θ)² ÷ 2g above the launch point. A firework shell launched at 70 m/s and 75° (g = 9.8 m/s²) climbs 233 m in 6.90 s (OpenStax Example 4.7).

### Does mass change projectile motion?

Not without air resistance: every mass falls with the same acceleration g, so the path depends only on the speed, angle, height and gravity.

### Why is the real range shorter?

Air resistance slows the projectile, so a real ball or shell lands short of the ideal range, often by a lot at high speeds. The page leaves air resistance out.

## Sources

- OpenStax, University Physics Volume 1, §4.3 Projectile Motion (equations of motion, maximum height, time of flight, range; Example 4.7). https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion (retrieved 2026-10-03)
- NIST SP 811, Appendix B.8 (standard acceleration of gravity 9.80665 m/s²; 1 ft = 0.3048 m; 1 mi = 1,609.344 m). https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b8 (retrieved 2026-10-03)