Projectile Motion Calculator

Physics of 2D Kinematics & Parabolic Trajectories

Projectile motion analyzes objects propelled into space subject only to the downward force of gravity (neglecting aerodynamic drag).

Independence of Perpendicular Motions

Galileo Galilei demonstrated that horizontal motion and vertical motion are completely independent:

• Horizontal Motion (X-axis): Moves at constant velocity), so x(t) = v_0x t.

• Vertical Motion (Y-axis): Undergoes constant gravitational acceleration), so y(t) = h_0 + v_0y t - (1) / (2) g t^

2.

Optimum Launch Angles

• Flat Ground): Maximum horizontal range occurs at exactly 45°.

• Elevated Cliff Launch (h_0 > 0): Maximum range occurs at an angle less than 45° (typically 35° to 42° depending on cliff height).

How to Use This Calculator

Enter your Initial Launch Velocity (v₀) (e.g. 25), Velocity Unit — choose from Meters per Second (m/s), Feet per Second (ft/s), Kilometers per Hour (km/h), Launch Angle (θ in degrees, 0° to 90°) (e.g. 45), Initial Launch Height (h₀ above ground) (e.g. 0 for flat ground), Height Unit — choose from Meters (m), Feet (ft), and Gravitational Acceleration (g) — choose from Earth (9.80665 m/s² / 32.174 ft/s²), Moon (1.62 m/s²), Mars (3.72 m/s²). Measure twice before entering dimensions — small measurement errors scale up significantly in material estimates.

The Formula

Classical 2D kinematics equations of parabolic projectile motion under constant acceleration (g): Components: v_0x = v_0 (θ), v_0y = v_0 (θ) to Trajectory Peak: t_apex = v_0yg = (v_0 (θ)) / (g) Altitude: H_ = h_0 + v_0y^22g = h_0 + (v_0^2 ^2(θ)) / (2g) Flight Time: t_flight = v_0y + √(v_0y)^2 + 2 g h_0g Range: R = v_0x × t_flight = (v_0 θ) × t_flight (For h_0 = 0: R = (v_0^2 (2θ)) / (g))

Worked Example

For example, launching a projectile at 25 m/s (55.9 mph) at an angle of 45° from ground level): . Peak height is . Total flight time is 3.605 seconds, and horizontal range is \9.80665 = 63.73 meters (209.09 ft).

Understanding Your Result

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