Find the best: where the slope is zero

Longest throw, biggest box, fastest rescue: slide to where the tangent goes flat.

Interactive

range (m)26.24

slope1.091 m per degree

best choice45

Problem

launch angle (°)20

The math, with your numbers

R(\theta) = \frac{v^2 \sin 2\theta}{g}, \quad v = 20\ \text{m/s}

R'(\theta) = \frac{2v^2\cos 2\theta}{g} = 0 \;\Rightarrow\; 2\theta = 90^\circ \;\Rightarrow\; \theta = 45^\circ

The slope says: go further right to do better.

Try this

→Longest throw: find the angle where the tangent is flat. Then compare 30° and 60°.
→Biggest box: what happens to the volume if you cut nothing, or cut 15 cm?
→Fastest rescue: at the best point, compare the two sin θ / v numbers.

Want to know why it behaves like this?

The lesson “Finding the Best: Optimization” explains it step by step: Why does 45° throw farthest, and why does light bend?

Find the best: where the slope is zero

Longest throw, biggest box, fastest rescue: slide to where the tangent goes flat.

Find the best: where the slope is zero

Interactive

range (m)26.24

slope1.091 m per degree

best choice45

Problem

launch angle (°)20

The math, with your numbers

R(\theta) = \frac{v^2 \sin 2\theta}{g}, \quad v = 20\ \text{m/s}

R'(\theta) = \frac{2v^2\cos 2\theta}{g} = 0 \;\Rightarrow\; 2\theta = 90^\circ \;\Rightarrow\; \theta = 45^\circ

The slope says: go further right to do better.

Try this

→Longest throw: find the angle where the tangent is flat. Then compare 30° and 60°.
→Biggest box: what happens to the volume if you cut nothing, or cut 15 cm?
→Fastest rescue: at the best point, compare the two sin θ / v numbers.

Want to know why it behaves like this?

The lesson “Finding the Best: Optimization” explains it step by step: Why does 45° throw farthest, and why does light bend?