Skip to content

The shape of a throw

Gravity and orbits · level 3, lesson 4 of 6

A question to start

You want to throw a ball as far as you can across a field. Should you throw it flat, straight up, or somewhere in between?

The idea

A thrown ball does two things at once:

  • It moves across at a steady speed (if the air does not slow it much).
  • It moves up and down like a ball thrown straight up: slowing on the way up, stopping, speeding up on the way down.

Put the two together and you get a smooth arch called a parabola. Every thrown ball, every jet from a fountain and every jumping frog follows one.

20°45°: farthest70°

At the same throwing speed:

  • Thrown flat, the ball moves across fast but lands almost at once.
  • Thrown straight up, it stays in the air a long time but goes nowhere across.
  • In between is best. Without air, the farthest throw is at 45°, halfway between flat and straight up. Throws at 30° and 60° land at the same spot, a bit short of the 45° one.

On Mars, gravity pulls about a third as hard, so the ball stays up longer and keeps moving across for longer. The same throw goes about 2.6 times as far. A ball you can throw 30 m at home would go nearly 80 m on Mars.

Worked examples

1. A long jumper. Long jumpers take off at about 20°, not 45°: they run so fast that they cannot push up hard enough for a steep jump. They still trace a parabola.

2. A garden hose. Point the hose flat, then at 45°, then nearly straight up, without changing the tap. The 45° jet lands farthest.

Try it

  1. What is the shape of a throw called?
  2. At what angle does a throw go farthest, without air?
  3. Two throws at the same speed, at 30° and 60°. Which goes farther?
  4. A ball you throw 20 m on Earth: about how far on Mars?
Answers
  1. A parabola.
  2. 45°.
  3. They land at the same distance (without air). The 60° one goes higher.
  4. About \(20 \times 2.6 = 52\) m.

Common mistake

"The ball flies straight, then drops." Cartoon characters run off cliffs and then fall. Real balls start curving the moment they leave your hand, because gravity pulls on them all the time.

Where it leads