Projectile Motion
Loading simulation…
flagWhat you'll discover
- arrow_forwardSplit projectile motion into independent horizontal and vertical parts
- arrow_forwardPredict range, maximum height and flight time from launch speed and angle
- arrow_forwardFind the optimum launch angle by experiment
- arrow_forwardSee how air resistance shortens and distorts the ideal parabola
Two motions in one
The secret of projectile motion is that the horizontal and vertical motions are completely independent. Horizontally, with no air resistance, nothing pushes or pulls the projectile, so it moves at constant velocity vₓ = v cos θ. Vertically, gravity decelerates it on the way up and accelerates it on the way down: v_y = v sin θ − gt.
Combine a steady drift sideways with a uniformly accelerated rise-and-fall and you get a parabola. Galileo proved this in the 1600s, and the same mathematics guides every basketball shot, artillery shell and water fountain.
Range, height and time
Three formulas summarise the ideal flight over level ground. Flight time: t = 2v sin θ / g, set by the vertical motion alone. Maximum height: H = v² sin²θ / 2g, reached when the vertical velocity hits zero. Range: R = v² sin 2θ / g.
The range formula contains sin 2θ, which peaks when 2θ = 90°, that is θ = 45°. Notice also that complementary angles such as 30° and 60° give exactly the same range — one shot flies low and fast, the other high and slow, but they land in the same spot.
What air resistance changes
Real air pushes back against motion with a drag force that grows roughly with the square of speed. Drag steals horizontal speed throughout the flight, so the projectile falls more steeply than it rose: the path becomes a lopsided curve, shorter and lower than the ideal parabola.
With drag switched on, the best launch angle drops below 45° — typically 30° to 40° depending on speed — because spending less time in the air means losing less speed to drag. This is why shot-putters and footballers release below 45° in practice.