How projectile motion is modelled
Once a projectile leaves your hand or a launcher, the only force acting on it (in this idealised model) is gravity pulling straight down. That lets you treat the horizontal and vertical motion as two independent problems: the sideways speed stays constant, while the upward speed slows, stops at the peak, then reverses.
Splitting the launch speed into its horizontal part (v·cosθ) and vertical part (v·sinθ) is the key step. The flight time comes entirely from the vertical motion — how long it takes gravity to bring the upward speed back to its mirror image on the way down — and the range is simply the constant horizontal speed multiplied by that time.
Reading the results and the trajectory
The range is how far the projectile travels horizontally before returning to launch height, the maximum height is the peak of the arc, and the flight time is how long it spends in the air. The trajectory chart plots height against horizontal distance, tracing the familiar parabola.
A 45° launch gives the greatest range over level ground, because it balances staying airborne (which favours a steep angle) against covering ground quickly (which favours a shallow one). Angles equally spaced above and below 45° produce the same range but different peak heights.
Assumptions and limitations
This is the textbook, drag-free model. It assumes launch and landing happen at the same height, ignores air resistance, spin and wind, and uses a constant g of 9.81 m/s². For light or fast objects — a feather, a golf ball, an arrow — air resistance shortens the real range noticeably, so treat these figures as an idealised upper bound rather than a field measurement.
Formula
range = v²·sin(2θ)/g; maxHeight = (v·sinθ)²/(2g); time = 2·v·sinθ/g