What your result means
Range is the horizontal distance covered before the projectile returns to ground level. Maximum height is the apex of the arc measured from the ground, not from the launch point — so if you launch from a platform, the apex already includes that head start.
Flight time is how long the projectile is airborne, and impact speed is how fast it is travelling the instant it lands. Launch and land at the same height and the impact speed matches the launch speed exactly; launch from higher up and it lands faster, because gravity has pulled it through an extra drop. If you want that speed in other units, the speed converter will turn m/s into mph or km/h.
Why this one is different
Range, apex, flight time and impact speed come from the same launch, so a change to the angle can be judged on all four at once instead of one at a time. Where the inputs push the model past what it can honestly represent, a warning says so rather than returning a trajectory that quietly ignores the air.
How it works
Projectile motion splits cleanly into two independent problems. Horizontally there is no force acting, so the projectile keeps its starting horizontal speed for the whole flight. Vertically, gravity pulls down at a constant rate, so the vertical speed falls steadily, reaches zero at the apex, then grows again on the way down.
Because the two axes never interfere, the path is a parabola, and the whole flight is fixed the moment the projectile leaves the launcher. That independence is also why a bullet fired horizontally and a bullet dropped from the same height hit the ground at the same time. The energy side of the same story is covered by the kinetic energy calculator.
How to use this simulator
- Set the launch speed — how fast the projectile leaves the launcher.
- Set the angle above the horizontal. Try 45° first, then compare 30° and 60°: they give the same range from ground level.
- Raise the launch height to throw from a cliff or a rooftop rather than the ground.
- Switch gravity to the Moon or Mars to see the same throw stretch out.
- Press play to animate, or read the four figures straight off the panel — they update as you drag.
Formula
vy = v · sin θ
tflight = ( vy + √( vy² + 2 g h ) ) ÷ g
range = vx · tflight
apex = h + vy² ÷ ( 2 g )
vimpact = √( v² + 2 g h )
- v — launch speed, in metres per second
- θ — launch angle above the horizontal, in degrees
- h — launch height above the ground, in metres
- g — gravitational field strength, in m/s² (9.80665 on Earth)
Example calculation
Throw at 20 m/s at 45° from ground level on Earth:
vx = 20 × cos 45° = 14.1421 m/s
tflight = 2 × 14.1421 ÷ 9.80665 = 2.88 s
range = 14.1421 × 2.8842 = 40.79 m
apex = 14.1421² ÷ ( 2 × 9.80665 ) = 10.20 m
vimpact = √( 20² + 0 ) = 20.00 m/s
Set the controls to those values and the panel shows the same four figures. Switch gravity to the Moon and the identical throw carries 246.15 m instead of 40.79 m.
Frequently asked questions
Why is 45 degrees the best launch angle?
When the projectile starts and lands at the same height and air resistance is ignored, range depends on sin(2 × angle), which peaks at 45°. Launch from above ground level and the ideal angle drops below 45, because extra falling time rewards a flatter, faster shot.
Does this simulator include air resistance?
No. It models motion in a vacuum, so horizontal velocity stays constant and the path is a perfect parabola. Real projectiles fall short of these numbers, and the gap widens with speed and with light objects such as a shuttlecock or table tennis ball.
Why does the impact speed equal the launch speed?
With no air resistance and equal start and finish heights, energy is conserved, so the projectile lands at the speed it left with. Launch from a height and it lands faster, because gravity has done extra work over the additional drop.
Related calculators
Assumptions & limitations
- No air resistance. This is vacuum motion. A real thrown ball falls short of the range shown here, and a light object falls far short.
- The projectile is a point. Spin, lift and shape are ignored, so effects like a curving football or a lofted golf ball are outside the model.
- Gravity is uniform. Field strength is treated as constant, which is safe for the heights here but not for orbital distances.
- The ground is flat and level. Landing is measured at height zero, so slopes and uneven terrain are not modelled.
- The arc is drawn to a single uniform scale. Horizontal and vertical use the same metres-per-pixel, so the shape you see is the true shape rather than a stretched one.
Further reading
Our guide to force, work, energy and power covers how these quantities connect, and why energy conservation explains the impact speed this simulator reports.
Sources & references
The constants and the equations of motion used on this page come from the following. These are specific to projectile motion and gravitational field strength.
- NIST — standard acceleration of gravity, the exact 9.80665 m/s² used as the Earth value
- NASA NSSDC — Moon fact sheet, surface gravity 1.62 m/s²
- NASA NSSDC — Mars fact sheet, surface gravity 3.71 m/s²
- OpenStax College Physics — projectile motion, the derivation behind the formulas above