Basketball Shot

Can you sink a free throw? Choose your launch angle and speed and shoot at a real-size hoop, 3.05 m high.

Shoot from
Air drag
Result
Take a shot
Flight time
0.00 s
Highest point
2.10 m
Speed now
7.2 m/s

The maths, live

Split the launch vx = v cos θ, vy = v sin θvx = 4.43 m/s, vy = 5.67 m/s
Height at the hoop y = y₀ + x tan θ − g x² ÷ (2v² cos² θ)y = 3.06 m (rim is 3.05 m)
Speed needed v = √(g x² ÷ (2cos² θ (x tan θ − Δy)))v = 7.20 m/s at 52°

A higher arc drops into the hoop more steeply, so it has a bigger target. Try 45° and 55° from the free-throw line. (The maths here leaves out air drag.)

How it works

Once the ball leaves your hands, the only big force on it is gravity, pulling it down. Its sideways speed stays the same, while its upward speed shrinks, then turns into a downward speed. Together they make a curved path called a parabola.

Split the launch speed into a sideways part (v cos θ) and an upward part (v sin θ) to work out where the ball goes. A higher arc drops into the hoop more steeply, so the hoop looks bigger to the ball. Turn on air drag to see how real air slows the ball a little.

Try this

  1. From the free-throw line, find the speed that swishes at 52°.
  2. Find two different angles that both score. Which one is easier to get right?
  3. Turn on air drag. How much faster do you need to throw?