Range, apex, and the best angle

Every launch asks the same three questions: how high, how long, how far. All three answers come from the split you already know — vertical decides the clock, horizontal spends it.

The three answers

Launching at speed v₀ and angle θ from flat ground (so v_x = v₀cos θ, v_y = v₀sin θ at t = 0):

Multiply out the range and a double-angle identity from trig appears:

$R = \frac{v_0^2 \sin(2\theta)}{g}$

The best angle

sin(2θ) peaks when 2θ = 90° — so 45° maximizes range on flat ground. And because sin(2 · 30°) = sin(2 · 60°), complementary angles land on the same spot:

At 45° the launch speed splits evenly between "go up" (which buys flight time) and "go forward" (which spends it). Tilt either way and you trade too much of one for the other.

What a launch height changes

Fire from a cliff and the flight lasts longer than the symmetric formula says — the projectile keeps falling past its launch level. The vertical equation becomes a full quadratic in t, and its positive root is the flight time. Two consequences worth knowing:

• Every answer still comes from "vertical sets the clock, horizontal spends it" — only the clock calculation grew.
• The best angle drops below 45° when launching from height (the higher the cliff, the flatter the optimal throw).

Test both claims in the projectile motion simulator — give it a launch height, then hunt for the range-maximizing angle with the slider.

Why this matters

This is the module's payoff: three formulas you can re-derive from a picture instead of memorize. And the next module answers the question these lessons kept postponing — why is the acceleration constant? Enter Newton.