Projectile Motion Simulator

How the numbers are computed

The simulator treats the two directions as separate problems, because that is exactly what the physics does. Nothing pushes the projectile horizontally after launch, so its horizontal velocity never changes. Gravity acts only vertically, pulling straight down with a constant acceleration of g = 9.8 m/s² on Earth (about 1.6 on the Moon and 3.7 on Mars). The horizontal and vertical motions share nothing but the clock.

A launch at speed v₀ and angle θ above the horizontal starts with the components v_0x = v₀ cos θ and v_0y = v₀ sin θ. When you drag the arrowhead you are editing those two numbers directly. From there the constant-acceleration equations produce the whole flight:

• Position. x = v_0x t, and y = h + v_0y t - (1/2) g t², where h is the launch height above the landing plane.
• Velocity. v_x stays equal to v_0x for the entire flight; v_y = v_0y - g t. The scrubber evaluates both at your chosen t and draws them to scale.
• Time of flight. Setting y = 0 gives a quadratic in t; the positive root is the moment of landing.
• Range and max height. Range is v_0x times the time of flight. Max height is h + v_0y²/(2g), the point where v_y passes through zero.

Why does 45° win on range only when the launch height is zero? Range rewards two things that fight each other: hang time, which favors steep angles, and horizontal speed, which favors shallow ones. Launching and landing at the same height, the trade balances exactly at 45°, and complementary angles such as 30° and 60° land in the same spot. Launch from a platform and the symmetry breaks: the height buys hang time for free, so a somewhat shallower angle, which spends more of v₀ on horizontal speed, carries farther. Raise h in the simulator and watch the best angle slide below 45°.