Projectile Motion Calculator

Projectile motion describes an object moving through space after launch while gravity provides the only acceleration. In the ideal model, horizontal acceleration is zero and vertical acceleration is constant and downward. Splitting the initial velocity into perpendicular components makes the motion straightforward: horizontal velocity is v cos θ and initial vertical velocity is v sin θ. The calculator combines those components with the launch height and gravitational acceleration to determine when the projectile returns to ground level. Flight time comes from solving the vertical position equation y = h + vyt - gt² / 2 for the nonnegative time at which y is zero. The positive quadratic root is used so launches above ground and launches with a downward component are handled correctly. Horizontal range is then horizontal velocity multiplied by flight time. Maximum height equals the initial height plus the vertical kinetic head vy² / 2g when the projectile initially travels upward. For horizontal or downward launches, the initial point is already the maximum height. Impact speed is calculated from the unchanged horizontal component and the final vertical component vy - gt. With no aerodynamic drag, energy conservation means a projectile landing at its launch elevation has the same speed magnitude that it had at launch. A lower landing elevation increases impact speed because gravitational potential energy becomes kinetic energy. The direction of impact is not displayed, but it can be found from the arctangent of final vertical velocity divided by horizontal velocity. The familiar maximum-range result of 45 degrees applies only when launch and landing elevations are equal, gravity is constant, and drag is absent. A raised launch point generally shifts the range-maximizing angle below 45 degrees. Real balls, vehicles, droplets, and other projectiles experience air resistance, wind, lift, spin, changing gravity, and terrain. Those effects can substantially alter a long or fast trajectory, so this ideal calculator is best for textbook kinematics, preliminary estimates, and controlled experiments where drag is small. Use meters, seconds, and meters per second throughout. Angles are measured from the horizontal: positive angles point upward and negative angles point downward. Standard Earth gravity is prefilled as 9.80665 meters per second squared, but local gravity varies slightly with latitude and elevation and can be changed. Gravity on another body can also be entered for comparison. The ground is defined as zero height, and initial height must not be negative. This calculator supports physics homework, laboratory planning, sports approximations, ballistics education, and quick trajectory checks. Always distinguish an ideal prediction from a safety envelope. Never use a drag-free point-mass result as the sole basis for operating machinery, launching objects, setting exclusion zones, or making decisions where an inaccurate landing point could cause harm.