How to Calculate Projectile Motion — Free Physics Calculator (2026)

Projectile motion describes the curved path of any object launched into the air under gravity alone. The free projectile motion calculator computes range, maximum height, time of flight, and velocity components from your initial conditions — no physics textbook required.

How to Use the Tool

1. Enter the initial launch velocity in metres per second (m/s) and the launch angle in degrees above the horizontal.
2. Optionally enter a launch height above the ground. Leave it at zero for ground-level launches.
3. The calculator instantly displays horizontal range, maximum height, total time of flight, and both velocity components at launch.

How It Works

At launch, the initial velocity v is split into two independent components using the angle θ:

• Horizontal: v_x = v · cos(θ) — constant throughout the flight (no air resistance)
• Vertical: v_y = v · sin(θ) — decreases due to gravity (g = 9.81 m/s²)

From these components the key results follow:

• Time of flight (ground launch): t = (2 · v · sin(θ)) ÷ g
• Maximum height: H = (v · sin(θ))² ÷ (2 · g)
• Horizontal range: R = v² · sin(2θ) ÷ g

When a non-zero launch height is provided, the time of flight is solved from the quadratic equation for vertical displacement, and the range is updated accordingly. Gravity is fixed at 9.81 m/s² (standard Earth surface value).

Use Cases

• Physics students verifying homework answers or exploring how changing launch angle and speed affect trajectory.
• Engineering applications such as sizing drainage channels, estimating debris fall zones, or designing catapult mechanisms.
• Sports science — analysing the optimal release angle for shot put, javelin, long jump, or basketball free throws.
• Game development — computing realistic ballistic trajectories for projectile weapons, grenades, or physics-based puzzles.
• Teaching demonstrations — instructors can use the tool live to show how range and height respond to parameter changes.

Frequently Asked Questions

What is the optimal launch angle for maximum range?

For a ground-level launch with no air resistance, the range formula R = v² · sin(2θ) ÷ g is maximised when sin(2θ) = 1, i.e. when 2θ = 90°. This gives θ = 45°. If the launch point is elevated above the landing point, the optimal angle drops below 45°; if the landing point is higher than the launch, the optimal angle exceeds 45°.

Does launch height affect the range?

Yes, significantly. A higher launch point gives the projectile more time in the air, which extends horizontal range even for the same initial velocity and angle. The calculator accounts for this: enter your launch height in the optional field and the range and time of flight will update to reflect it.

What units does the tool use?

All inputs and outputs use SI units: metres per second (m/s) for velocity, metres (m) for height and range, seconds (s) for time of flight, and degrees (°) for the launch angle. If your values are in other units — feet, miles per hour, kilometres per hour — convert them first. 1 mph ≈ 0.447 m/s; 1 ft ≈ 0.305 m.