How this instrument works
A projectile motion lab hands you one measurement pair — the launch speed off a spring gun or slingshot, and the angle read from its base — and expects three separate answers back: how long the shot stays up, how far it lands, and how high it peaks. All three trace back to splitting that single velocity into a horizontal piece, v·cosθ, that never changes, and a vertical piece, v·sinθ, that gravity erodes at a constant 9.80665 m/s² every second. Time of flight is twice the time that erosion takes to zero out the vertical piece; range is the horizontal piece multiplied by that same time; maximum height is what the vertical piece alone would climb before gravity cancels it.
Because all three outputs come from the same two inputs, they can check each other. Eliminate v·sinθ between the time-of-flight and height formulas and a compact identity falls out: maximum height equals g·t² ⁄ 8, using nothing but the time reading and standard gravity. A student who has already timed a shot with a photogate can verify the calculated peak height without touching the velocity or angle fields again — useful when the launcher's stated muzzle speed is a rated figure rather than something independently measured.
The three formulas assume launch and landing sit at the same height and that nothing but gravity acts on the shot after release. Neither assumption survives contact with a real lab bench: a tabletop launcher's muzzle usually sits a few centimetres above the surface the ball eventually lands on, and a light foam or plastic ball loses several percent of its predicted range to air drag before it gets there. A measured range that comes in consistently short of this calculator's number is not a mistake — it is the gap between the vacuum-and-level-ground case these equations solve and the room the experiment actually happens in.
- Enter Launch velocity in m/s — the launcher's rated muzzle speed, or a value measured with a photogate at the muzzle.
- Enter Launch angle in degrees above horizontal, read directly off the launcher's angle scale.
- Read Time of flight for the total seconds the shot stays airborne before landing.
- Read Horizontal range and Maximum height — both computed from the same velocity and angle you just entered.
- Cross-check Maximum height by hand with h_max = g·t² ⁄ 8, using only the Time of flight reading and standard gravity.
Worked example — a 20 m/s launch at 45°
Set Launch velocity to 20 m/s and Launch angle to 45°, the classic lab-bench setup for a spring launcher check. Time of flight comes out to 2 × 20 × sin(45°) ⁄ 9.80665 = 2.884193 s. Horizontal range follows from 20² × sin(90°) ⁄ 9.80665 = 40.788649 m, and Maximum height works out to (20 × sin 45°)² ⁄ (2 × 9.80665) = 10.197162 m — the full trajectory summary a lab write-up needs from one pair of readings.
The cross-check confirms it: g·t² ⁄ 8 = 9.80665 × 2.884193² ⁄ 8 also comes to 10.197162 m, matching Maximum height exactly using only the Time of flight reading. That agreement is what a lab report is actually graded on — not that the numbers look plausible, but that two independent routes through the same data land on the identical figure to six decimal places.
Questions
Why does this calculator give time, range, and height together instead of one at a time?
Because a projectile motion lab report needs all three from a single measurement pair — launch velocity and launch angle — and checking them against each other catches transcription errors that a lone range or height calculator would never surface. Type the two inputs once and Time of flight, Horizontal range, and Maximum height all update together.
How do I check Maximum height without re-entering velocity or angle?
Use h_max = g·t² ⁄ 8, an identity built by eliminating v·sinθ between the height and time-of-flight formulas. Take the Time of flight reading, square it, multiply by standard gravity, 9.80665 m/s², and divide by eight; for the 45° example on this page that gives 10.197162 m, matching Maximum height exactly.
Why does my measured range fall short of the Horizontal range this calculator reports?
Two lab-bench realities the formula ignores: the launcher's muzzle usually sits a few centimetres above the table the ball lands on, which shortens the actual fall path, and a light foam or plastic ball sheds a noticeable share of its range to air drag. A measured value running several percent under the calculated one is expected, not a sign of a bad reading.
Does Time of flight always get longer as I increase Launch angle?
Yes, all the way to 90°, since Time of flight depends only on sinθ, which climbs steadily from 0 to 1 across that range. Horizontal range behaves differently — it peaks at 45° and falls again beyond it — so a steeper shot always hangs in the air longer, even past the angle that maximizes distance.
Do I need to enter Launch angle in degrees?
Yes, degrees measured up from horizontal is what the field expects, matching the scale printed on most classroom projectile launchers: 0° fires dead level, 90° fires straight up. Feeding in a radian value by mistake, 0.785 instead of 45, produces numbers nowhere close to a real trajectory.