Projectile Motion: Range, Height and Time Without Drag

Throw a ball and physics writes the story for you. Up to the apex, over the top, down to the grass — and every number along the way follows from just two inputs: how fast you launched and at what angle. The ideal-projectile equations are most students' first meeting with superposition, and they remain the working model for fountains and stunt jumps alike, right up until air resistance gets a vote.

Two independent motions, one trajectory

The launch velocity splits into a horizontal and a vertical part:

vx = v · cos θ   (constant — nothing touches it)
vy = v · sin θ   (decays at g, reverses, returns)

Gravity pulls straight down at 9.81 m/s² and never sideways. That one-sided arrangement is the whole trick: the horizontal motion is uniform, the vertical motion is simple free fall, and the path — a parabola — is just what you get when you plot a steadily advancing x against an up-then-down y. The two motions do not talk to each other. The ball keeps moving forward at launch speed's horizontal share even at the moment it stops rising.

The three headline numbers

Because the vertical trip is symmetric — the time rising equals the time falling — everything of interest follows from vy and g:

time of flight  t  = 2·v·sin θ / g
max height      H  = (v·sin θ)² / (2g)
range           R  = v² · sin(2θ) / g

Check them against a textbook case: 20 m/s at 45° gives a flight time of 2.88 s, a maximum height of 10.19 m, and a range of v²/g = 400/9.81 ≈ 40.77 m — the sin(2θ) term equals 1 at exactly 45°, which is why that angle maximises distance on level ground. At 30° and 10 m/s, the range is 8.83 m; at 60° and the same speed, exactly the same 8.83 m, because sin(60°) = sin(120°). Complementary angles always tie: one buys hang time, the other buys speed, and at 45° the trade is optimal.

Height is the vertical component's kinetic energy converted to potential energy at the top. A 90° launch — straight up — sends every bit of speed vertical: range zero, height v²/2g, and a flight time of 2v/g. The projectile returns to the launch point as fast as it left, which is not a paradox but energy conservation with nothing to lose it to.

The landing mirrors the launch

With no drag, mechanical energy is conserved: whatever kinetic energy the climb removed, the fall restores. So the impact speed equals the launch speed, the horizontal velocity is unchanged, and the final vertical velocity equals the initial one in magnitude but points down. The landing angle is the mirror image of the launch angle. On flat ground with no drag, a projectile that lands slower than it launched means your simulation is leaking energy.

Where the ideal model breaks

Three assumptions do all the work here: no air resistance, flat launch and landing heights, and a uniform gravitational field. Each fails in instructive ways. Drag scales with speed squared and is merciless to light objects — a baseball loses a meaningful fraction of its ideal range, and a shuttlecock barely pretends to be a parabola. Uneven ground matters just as much: throw from a height and the projectile keeps falling past the symmetric return point, lengthening the flight, stretching the range, and shifting the optimal launch angle below 45°.

Speed and distance in these formulas are metres and metres per second; if your data is in miles per hour or feet, convert first so the units agree with g = 9.81 m/s². The Speed Converter and Length Converter handle that bookkeeping in one click.

Why this model is still worth learning

Every trajectory simulation, game engine and ballistics table starts by deciding how much of this ideal physics to keep. The ideal equations are the zeroth-order answer — good enough for slow, dense, short flights, and the correct baseline against which drag corrections are measured. They are also the cleanest illustration of vector decomposition in all of introductory physics: split the problem into independent directions, solve each, and reassemble. The ideal gas guide applies the same conservation logic to molecules instead of cannonballs.

Try it yourself

The Projectile Motion Calculator runs the exact equations above: enter launch speed and angle (and optionally switch gravity to the Moon or Mars) to get range, maximum height, flight time, component velocities and the impact state, all rounded sensibly and computed entirely in your browser. To convert your launch speed or read the range in feet, the Speed Converter and Energy Converter sit alongside it — the vertical-only case of this math is plain free fall, the same equations with the horizontal part switched off.