Range and angle.

Step 1 of 4

Hit the flag

There is a flag planted in the field. Your goal: launch the ball so that it lands on it. You have two controls, Launch angle and Launch speed, and the Scene marks where the ball landed.

Step 2 of 4

Which angle goes farthest

At a fixed speed, every angle shares that speed out differently between moving across and moving up. A low angle travels across quickly but lands soon; a high angle stays in the air a long time but makes slow progress.

Predict which angle wins, then watch.

Step 3 of 4

The range formula

The flight time is set by the vertical part: the ball goes up and comes back down in $t = \dfrac{2 v \sin\theta}{g}$$t =$$\dfrac{2 v \sin\theta}{g}$⁠. During that time it moves across at a constant speed $v\cos\theta$⁠, so:

$R = v\cos\theta \cdot \frac{2 v\sin\theta}{g} = \frac{v^2 \sin 2\theta}{g}$$R =$$v\cos\theta \cdot \frac{2 v\sin\theta}{g} =$$\frac{v^2 \sin 2\theta}{g}$⁠

$\sin 2\theta$⁠ is largest at $\theta = 45^\circ$⁠. Angles that add up to $90^\circ$⁠, such as $30^\circ$⁠ and $60^\circ$⁠, give the same range.

The angle and the speed are now fixed. Work out the range yourself.

Step 4 of 4

Double the speed

In the formula, the range depends on $v^2$⁠. Before you try it in the Scene, answer: what happens to the range when you double the speed?