Two motions at once
Throw a ball and it traces a smooth arc through the air. It looks complicated, but here's the secret that unlocks it — first noticed by Galileo:
A projectile is doing two completely separate, simple motions at the same time.
- Sideways: nothing pushes it sideways (ignoring air), so it moves at a constant velocity. Same distance every second.
- Up and down: gravity pulls it down, so it's in free fall — exactly like the last lesson. It slows on the way up, stops at the top, and speeds up on the way down.
Put these two together and you get the curve. Watch the two arrows on the ball in the simulation.
A cannon on flat ground that launches a ball toward a target. Set the launch angle and speed and choose Earth, the Moon or Mars, then press Fire. The ball follows a parabola; the readings show range, maximum height, flight time and how many shots hit the target.
Cannon range
InteractivePlanet
The math, with your numbers
Try this
- →Hit the red flag. Then find a second, different angle that hits it too!
- →With the speed fixed, which angle gives the longest range?
- →Watch the blue vx arrow during flight. Does it ever change length? What about the amber vy arrow?
Why don't the two motions interfere?
Make a prediction first
You drop one ball and, at the same moment, fire another one horizontally from the same height. Which hits the floor first?
Think about it, then click to reveal the answer.
Make a prediction first
You drop one ball and, at the same moment, fire another one horizontally from the same height. Which hits the floor first?
Think about it, then click to reveal the answer.
They land at the same time! The horizontal speed doesn't help or hinder the falling at all. Both balls start with zero vertical velocity and both feel the same gravity, so they fall in exactly the same way. The fired ball just lands farther away.
The math of a launch
Split the launch velocity into its components using the angle (exactly as in the Vectors lesson):
Say it out loud: “sideways speed is v times cos theta; starting upward speed is v times sin theta.”
Launch speed· measured in m/s
How fast it leaves the cannon.
Launch angle· measured in degrees
How far above horizontal you aim.
Sideways velocity· measured in m/s
Stays the same the whole flight.
Starting upward velocity· measured in m/s
Gravity eats away 9.8 m/s of this every second.
Worked example: kick a ball at 20 m/s, 30° above the ground
Step 1 of 4- 1Split the velocity. (cos 30° ≈ 0.866, sin 30° = 0.5.)
Combining these steps into one formula gives the famous range equation (for flat ground, no air):
Wrap-up
Key ideas to remember
- A projectile = constant-velocity sideways motion + free fall vertically, happening at the same time.
- The two directions are independent: gravity only affects the vertical part.
- The path is a parabola. Flight time comes from the vertical motion; range = .
- On flat ground, 45° gives the maximum range, and complementary angles (like 30° and 60°) tie.
Check your understanding
Sources & further reading
Everything in this lesson agrees with these references. They're all free to read.