Projectile motion

How it works

Without air resistance the motion combines constant velocity horizontally with constant acceleration vertically, so the path is a parabola. Launched from the ground, the range is d = v₀²·sin 2α / g: it is greatest at 45°, and 30° and 60° give the same range. Air resistance grows with the square of the speed, shortens the range and moves the best angle below 45°.

x = v₀·cos α·ty = h₀ + v₀·sin α·t − ½·g·t²d = v₀²·sin 2α / g

Try it yourself

1. With the defaults (45°, 40 m/s), keep the path with “＋ Add to comparison”. The range is 163.1 m.
2. Fire at 30° and then at 60°. Predict first: the range is the same (141.3 m); only the height and flight time differ.
3. Switch the environment to the Moon. Gravity is about six times weaker, so the range is about six times longer (987.7 m).
4. Back on Earth, turn on “Air resistance”. At 45° the range drops to 117.6 m, and the ball flies furthest at about 42°. Drag cannot be turned on for the Moon because it has no atmosphere; Mars has air about 60 times thinner.

Model limitations

The 12 cm ball is treated as a point with quadratic air drag (Cᴅ = 0.47). The model ignores spin, lift and surface curvature, and the ball lands on flat ground. It is not a tool for real ballistics.

Related experiments

• Levers and torque
• Gravity and orbits
• Double pendulum