Projectile Motion Calculator
Compute the range, maximum height, and flight time of a projectile launched at any angle and initial height. Ideal for physics students, engineers, and sports analysts studying ballistic trajectories.
About this calculator
Projectile motion splits an object's path into independent horizontal and vertical components. Horizontally, velocity is constant (ignoring air resistance); vertically, gravity decelerates the object. The horizontal range R when launched from height h is: R = (v²·sin(2θ))/g + (v·sin(θ)·√((v·sin(θ))² + 2·g·h))/g. Maximum height is H = h + (v·sin(θ))²/(2g). Flight time is T = (v·sin(θ) + √((v·sin(θ))² + 2·g·h))/g. These equations assume constant gravitational acceleration and no air drag. Angle θ is converted from degrees to radians before computing the sine and cosine terms.
How to use
Suppose a ball is kicked at v = 20 m/s, angle θ = 45°, from height h = 0 m, with g = 9.81 m/s². Step 1 – compute v·sin(45°) = 20 × 0.7071 = 14.14 m/s. Step 2 – sin(90°) = 1, so the first term = (400 × 1)/9.81 = 40.77 m. Step 3 – with h = 0, the square root term = 14.14, so the second term = (14.14 × 14.14)/9.81 = 20.39 m. Step 4 – total range R = 40.77 + 20.39 ≈ 40.77 m (second term vanishes when h = 0, confirming R = v²·sin(2θ)/g ≈ 40.77 m).
Frequently asked questions
What launch angle gives maximum range for projectile motion?
For a projectile launched from ground level (h = 0), the optimal angle is exactly 45°, which maximises sin(2θ) = 1. When the launch height is greater than zero, the optimal angle shifts below 45° because the extra altitude effectively extends flight time. You can experiment with this calculator by varying the angle while keeping other inputs fixed to find the true maximum range for your scenario.
How does initial height affect projectile range and flight time?
A higher launch point gives the projectile more time in the air before it reaches the ground, increasing both range and total flight time. The formula accounts for this through the term 2·g·h inside the square root; as h grows, so does the square root, adding extra horizontal distance. Even a modest elevation—like throwing from a 2 m platform versus ground level—can meaningfully extend range at low angles.
Why is gravitational acceleration set to 9.81 m/s² in projectile calculations?
9.81 m/s² is the standard average gravitational acceleration at Earth's surface, defined by the International System of Units. In reality, g varies slightly with latitude (9.78 m/s² at the equator, 9.83 m/s² at the poles) and altitude. For most engineering and classroom problems the difference is negligible, but you can override this field to model projectiles on other planets—for example, use g = 3.72 m/s² for Mars.