Projectile Motion Calculator

Projectile motion describes the curved path an object follows when launched into the air and subject only to gravity (no air resistance). The horizontal velocity remains constant throughout, while the vertical velocity changes due to gravitational acceleration. The resulting path is a parabola.

The main equations are: horizontal position x(t) = v₀cosθ · t; vertical position y(t) = y₀ + v₀sinθ · t − ½g·t²; horizontal velocity vx = v₀cosθ (constant); vertical velocity vy(t) = v₀sinθ − g·t. Maximum height is yₘₐₓ = y₀ + (v₀sinθ)²/(2g) and time of flight is found by solving y(t) = 0.

When launched from ground level (initial height = 0), a projectile achieves maximum range at a 45° launch angle. However, if the launch height is above the target, the optimal angle is slightly less than 45°. This calculator lets you experiment with different angles to find the maximum range for your specific scenario.

No — projectile motion applies to any object moving under gravity after being launched, regardless of direction. A ball thrown straight up is a special case of projectile motion with a 90° angle and zero horizontal range. Even an object dropped vertically from height follows projectile motion principles.

Common examples include a thrown ball, an arrow shot from a bow, a kicked football, water from a hose, or a cliff diver. Any object that is given an initial velocity and then moves freely under gravity (ignoring air resistance) follows projectile motion equations.

In standard projectile motion, horizontal acceleration is zero (constant horizontal velocity), and vertical acceleration equals gravitational acceleration g downward (approximately 9.81 m/s² on Earth). This calculator lets you choose different gravitational values for other planets as well.

The horizontal range depends on the initial speed, launch angle, initial height above the landing point, and gravitational acceleration. A higher speed or greater initial height increases range. The launch angle determines how the speed is split between horizontal and vertical components, with 45° being optimal for flat ground launches.

Air resistance makes projectile equations significantly more complex, requiring drag coefficients, object mass, and cross-sectional area. The standard kinematic model without air resistance gives exact analytical solutions and is the foundation taught in physics courses. For many practical purposes (dense objects at moderate speeds), the no-resistance model gives a very good approximation.