Projectile Motion Calculator

The range of a projectile — the horizontal distance it travels before landing back at its launch height — is calculated with R = v0² · sin(2θ) / g, where v0 is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity (9.81 m/s² on Earth). For example, launching at 20 m/s at a 45° angle gives a range of 20² × sin(90°) / 9.81 ≈ 40.77 meters. Range is maximized at a 45° launch angle for any given speed, assuming the projectile lands at the same height it was launched from.

A 45° launch angle gives the maximum range for a projectile launched and landing at the same height, because sin(2θ) reaches its peak value of 1 when θ = 45° (since 2 × 45° = 90°). Any angle above or below 45° produces a shorter range at the same initial velocity — for instance, 30° and 60° produce identical ranges to each other, but both fall short of what 45° achieves, since sin(60°) = sin(120°).

Maximum height is found with H = v0² · sin²(θ) / (2g), where v0 is the initial velocity, θ is the launch angle, and g is gravitational acceleration. This formula comes from the vertical component of velocity (v0·sinθ) decelerating under gravity until it reaches zero at the peak. For a 20 m/s launch at 45°, the maximum height works out to about 10.19 meters — notice this is exactly one-quarter of the range for this specific case, though that relationship only holds at a 45° launch angle.

Time of flight is the total time the projectile spends in the air, calculated as T = 2 · v0 · sin(θ) / g. This doubles the time it takes to reach maximum height, since — assuming level ground — the projectile takes exactly as long to fall back down as it took to rise. At 20 m/s and 45°, the time of flight is about 2.88 seconds.

No — this calculator uses the standard idealized projectile motion equations, which assume no air resistance (drag), no wind, and a constant gravitational field. In reality, air resistance slows a projectile's horizontal velocity over time and reduces both range and maximum height compared to the idealized formulas, with the effect becoming more significant at higher speeds or with lighter, less aerodynamic objects. These equations are still an excellent approximation for physics coursework and for objects that are dense and compact enough that drag has minimal effect.

The formulas used here — R = v0²sin(2θ)/g, H = v0²sin²(θ)/(2g), and T = 2v0sinθ/g — all assume the projectile launches and lands at the same height, such as a ball thrown across a flat field. If the projectile is launched from an elevated position (like a cliff or a cannon on a platform) and lands lower, the actual range and time of flight will be greater than these formulas predict, since gravity has extra time to act after the projectile would have otherwise landed. Calculating trajectories with unequal launch and landing heights requires solving the full quadratic equation of motion for vertical displacement.