Projectile motion
Why does a ball curve?
Throw a ball forward. It rises, slows down, turns and falls back.
The path is not a straight line and not a circle. It is a curve called a parabola.
To see why, we split the motion into two simpler motions.
Splitting the velocity
The launch speed u at angle θ splits into two parts.
Sideways: u cos θ. Nothing pushes sideways, so this part never changes.
Upward: u sin θ. Gravity takes away 9.8 m/s of it every second.
At the top the upward part is zero, but the ball still moves sideways.
Try it: launch a projectile
- Drag the angle slider and press Launch.
- Turn on the partner angle to compare θ with 90° − θ.
- Which angle gives the longest range?
Quick check
Two balls are thrown with the same speed, one at 30° and one at 60°. Which one lands farther away?
- The 30° ball lands farther
- The 60° ball lands farther
- They land at the same spot
- It depends on the mass of each ball
Answer
C. Range is R = u² sin 2θ / g. For 30°, sin 60° = 0.866. For 60°, sin 120° = 0.866. Same value, so the same range. Mass does not appear in the formula at all.
Summary
- Time of flight:
T = 2u sin θ / g - Maximum height:
H = u² sin²θ / 2g - Horizontal range:
R = u² sin 2θ / g
- The sideways velocity u cos θ never changes, because nothing pushes sideways.
- The upward velocity drops by 9.8 m/s every second, reaches zero at the top, then points down.
- Put the two together and the path is a parabola. The longest range is at 45°.