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Classical Mechanics

Projectile Motion

An interactive study of a body in free flight under uniform gravity — no air resistance.
Trajectory Projectile & velocity Apex Hover or click the arc to read coordinates
Max height
–m
Range
–m
Flight time
–s
Time to apex
–s
x(t) = v₀·cos θ · t   |   y(t) = y₀ + v₀·sin θ · t − ½ · g · t²
i About this experiment — click to learn the physics ▼

What you're looking at

This is a projectile motion experiment: a body is launched from a height y₀ with an initial speed v₀ at an angle θ above the horizontal, and then moves freely under gravity alone. We assume no air resistance, so the only force acting after launch is the body's weight, pulling straight down. Drag the sliders or type values to change the launch, press Launch to animate the flight, and hover or click the arc to read the exact coordinates at any instant.

The key idea: motion splits in two

The single most important principle is that the horizontal and vertical motions are independent. Gravity acts only vertically, so it never changes the horizontal velocity. We can therefore treat the launch velocity as two separate pieces and solve each direction on its own:

vₓ = v₀·cos θ horizontal velocity — constant
v_y = v₀·sin θ − g·t vertical velocity — changes with gravity

Horizontally the body moves at a constant speed (no horizontal force). Vertically it behaves like a ball thrown straight up: it decelerates, stops momentarily at the top, then accelerates back down at the gravitational acceleration g.

Equations of motion

Integrating those velocities over time gives the position at any moment t:

x(t) = v₀·cos θ · t uniform horizontal motion
y(t) = y₀ + v₀·sin θ · t − ½·g·t² uniform vertical acceleration

Eliminating t between these two shows the path is a parabola — the curved arc you see traced on screen.

Where the readouts come from

  • Time to apex: the body rises until v_y = 0, which happens at t = v₀·sin θ / g.
  • Maximum height: the highest point of the arc, H = y₀ + (v₀·sin θ)² / 2g.
  • Flight time: found by solving y(t) = 0 (when it lands), which accounts for the launch height.
  • Range: the horizontal distance covered during the whole flight, R = vₓ × flight time. For a launch from the ground this peaks at θ = 45°.

Things to try

Lower the gravity to the Moon (1.62 m/s²) and watch the same launch fly far higher and longer. Set complementary angles such as 30° and 60° at the same speed — from ground level they produce the same range. Add a launch height and notice the optimal angle drops below 45°.