π― Projectile Motion Calculator
Calculate range, maximum height, and time of flight from initial velocity and angle.
How to Use This Calculator
This calculator solves projectile motion for a launch and landing at the same height. Enter the initial speed and launch angle to get the range, maximum height, and flight time.
Enter the initial velocity in meters per second. This is the speed at the instant of launch.
Enter the launch angle in degrees above the horizontal. Use 45 for maximum range.
Click Calculate. You get the horizontal range, the peak height, and the total time of flight.
Remember the assumptions: no air resistance, flat ground, and constant g = 9.81 m/s^2.
Projectile Motion Formulas
A projectile is two independent motions sharing one clock. Horizontally, nothing pushes it, so horizontal velocity stays constant and distance is simply speed times time. Vertically, gravity accelerates it downward at g, so the vertical motion is a free-fall problem starting with an upward kick of v0 sinΞΈ. The launch angle splits the initial velocity between these two motions: cosine goes horizontal, sine goes vertical.
The famous 45-degree optimum falls out of sin(2ΞΈ), which peaks at 90 degrees. Steeper than 45 wastes velocity on height you do not need; shallower wastes it on a flight too short to use the time well. The symmetry is elegant: going up takes exactly as long as coming down, so the peak sits at the midpoint of the flight.
Worked Example
Where This Comes Up
Sports are projectile problems in disguise. A basketball free throw, a golf drive, a long jump, and a shot put all trade launch speed against angle, and coaches use exactly this math to find optimal release conditions. Fireworks designers time shell bursts from flight-time calculations, and game engines fake all of it with these same three formulas running every frame.
The trap is applying level-ground formulas to uneven terrain: a golf ball landing downhill flies farther than the range formula says. And at real sports speeds, air resistance is not negligible: a golf ball's dimples exist precisely because drag dominates its flight. Use this calculator for the ideal physics, then expect reality to shave something off.
Frequently Asked Questions
What launch angle gives the maximum range?
45 degrees, when launch and landing heights are equal and air resistance is ignored. The sin(2ΞΈ) term peaks at 2ΞΈ = 90 degrees.
Do 30 degree and 60 degree launches land at the same distance?
Yes. Any two angles that add to 90 degrees give the same range, because sin(2ΞΈ) = sin(180Β° - 2ΞΈ). The higher angle flies longer and peaks higher; the lower one gets there faster and flatter.
Why is air resistance ignored?
Drag depends on speed, shape, and air density, which turns clean algebra into differential equations. The vacuum model teaches the core physics and is accurate for dense, slow projectiles like shot puts.
What if the landing height differs from the launch height?
These formulas assume level ground. For a cliff or a raised platform, solve the vertical motion separately with the quadratic formula to find the true flight time.
Which matters more for range, speed or angle?
Speed, by far. Range scales with the square of initial velocity but only with sin(2ΞΈ) for angle. A 10 percent speed boost beats any angle tweak.