A basketball arcing toward the hoop, a golf ball off the tee, a water balloon lobbed over a fence — all three follow the same curved path, called a parabola, and that path is entirely predictable once you know the launch speed, angle, and starting height. This is projectile motion, one of the oldest solved problems in physics.
How to Use the Projectile Motion Calculator
Enter the launch velocity in meters per second, the launch angle in degrees above horizontal, and the initial height in meters (use 0 for ground level). The calculator returns the range — how far the projectile travels horizontally before landing — the maximum height it reaches, and the total time of flight.
The Two Motions Are Independent
The deepest insight in projectile motion, first worked out rigorously by Galileo, is that horizontal and vertical motion don't interact — gravity only ever acts vertically, so it never slows down or speeds up the horizontal component. A bullet fired perfectly horizontally and a bullet simply dropped from the same height at the same instant hit the ground at exactly the same time, even though one has traveled hundreds of meters sideways and the other has traveled zero. That's a genuinely surprising result the first time you see it demonstrated, and it's the core reason the horizontal and vertical formulas above can be solved completely separately from each other.
The Projectile Motion Formulas
A launch velocity splits into two independent components: a horizontal component that never changes (ignoring air resistance) and a vertical component that gravity constantly slows, stops, and reverses.
Horizontal speed: vₓ = v × cos(θ)
Vertical speed: v_y = v × sin(θ)
Time of flight: t = [v_y + √(v_y² + 2gh)] / g
Maximum height: h_max = h + v_y² / (2g)
Range: R = vₓ × t
Here v is launch speed, θ is launch angle, h is initial height, and g is standard gravity, 9.80665 m/s². Using the calculator's default example — 20 m/s at 45 degrees from ground level — horizontal and vertical speeds are equal (since sin and cos of 45° are both about 0.707), giving vₓ = v_y ≈ 14.14 m/s. Time of flight comes to roughly 2.88 seconds, maximum height to about 10.2 meters, and range to about 40.8 meters.
What This Calculator Doesn't Model
Like the free-fall calculator, this tool ignores air resistance, which matters more for projectiles than most people expect — a well-hit golf ball actually travels farther than a vacuum calculation would predict, thanks to backspin-induced lift, while a skydiver or a badminton shuttlecock is dominated by drag almost immediately. Treat these results as the idealized physics-textbook answer, accurate for dense, compact, slow-to-moderate-speed objects like a thrown ball, and increasingly approximate for anything light, fast, or aerodynamically shaped.
Why 45 Degrees Maximizes Range (on Level Ground)
When launching and landing at the same height, 45 degrees gives the longest possible range for a given speed — it's the angle that best balances "staying in the air long enough" against "moving forward fast enough." Steeper angles trade horizontal speed for more hang time; shallower angles do the opposite, and 45 degrees splits the difference exactly. This is why shot-putters and long jumpers train to release at angles near 40-45 degrees, and why a mortar aimed straight up (90 degrees) lands right back where it started, with zero range at all.
Launching From a Height Changes the Optimal Angle
The 45-degree rule only holds when launch and landing heights match. Throwing from a height — off a cliff, from a balcony, or a golfer's tee slightly above the fairway — favors a shallower angle than 45 degrees, because the extra fall time from height rewards more horizontal speed relative to hang time. This calculator's initial-height input lets you see that shift directly: increase the height while holding velocity and angle fixed, and you'll see the range grow even without changing the angle, because gravity now has extra distance to accelerate the object before it lands.
