Why 45° Throws the Farthest
By Lucian — builder & engineer, LK Forge
Ignore the air for a moment and a thrown ball's whole future is fixed by two numbers: how fast it leaves your hand and at what angle. The angle alone decides how the flight trades distance for height — and it hides a neat symmetry, because two different angles can drop the ball in exactly the same spot. Here is the range at every launch angle, worked out exactly — and then measured again with real air resistance, where the tidy 45° answer stops being true.
A single launch in the projectile-motion simulator. The arrow is the velocity — steep at launch, flattening as it approaches the top, then steepening again on the way down as gravity builds the downward speed back up.
| Launch angle | Range | Max height | Flight time |
|---|---|---|---|
| 15° | 45.9 m | 3.1 m | 1.58 s |
| 30° | 79.5 m | 11.5 m | 3.06 s |
| 45° ★ | 91.8 m | 23.0 m | 4.33 s |
| 60° | 79.5 m | 34.4 m | 5.30 s |
| 75° | 45.9 m | 42.8 m | 5.91 s |
Launch speed 30 m/s, g = 9.8 m/s², level ground, no air resistance. ★ marks the maximum range. Notice the range column is a mirror — 15° matches 75°, 30° matches 60° — while height and flight time climb straight up with the angle.
Two angles, one landing spot
The mirror in the range column is not a coincidence — it is built into the sine. A low, flat 30° shot spends little time in the air but nearly all of its speed is horizontal; a steep 60° shot keeps far more of its speed as height, so it hangs in the air more than twice as long, but crawls forward. The two effects trade off exactly, and both balls touch down 79.5 m away. What differs is everything else: the 60° shot climbs three times as high and stays up almost twice as long. If your goal is distance, 45° splits the difference perfectly; if it is hang-time or clearing a wall, steeper always wins.
The catch: real air moves the answer
Everything above assumes a vacuum. Add air resistance and the tidy picture bends: drag grows with the square of speed, so it bites hardest early in the flight when the ball is fastest, robbing the horizontal reach a flat launch depends on. The 45° optimum slips lower and the parabola turns lopsided. By how much, exactly? Rather than hand-wave, we switched drag on in the simulator and measured it.
We ran the simulator's own drag engine with no browser: the same fourth-order Runge–Kutta step it ships, integrating quadratic air resistance a = −K|v|·v − g (K = 0.006, step 1/240 s), launched from the ground and stopped at the interpolated landing point — exactly as the tool does. Then we swept the launch angle to find the true maximum, and swept the launch speed to see where it goes.
Range at 30 m/s. The vacuum curve is the textbook parabola peaking at 45°; the drag curve sits ~28% lower and peaks a little earlier, near 42.5° — a broad, flat optimum.
The drag-optimal angle falls steadily below the vacuum's 45° as the launch speed rises — because faster means more drag, and more drag punishes the slow, high part of the arc.
The optimum, speed by speed
| Launch speed | Best angle (drag) | Range there | Vacuum best (45°) | Range lost |
|---|---|---|---|---|
| 10 m/s | 44.8° | 9.7 m | 10.2 m | −4.5% |
| 20 m/s | 43.8° | 34.5 m | 40.8 m | −15.5% |
| 30 m/s | 42.5° | 65.8 m | 91.8 m | −28.4% |
| 40 m/s | 41° | 97.6 m | 163.3 m | −40.2% |
| 50 m/s | 39.8° | 127.5 m | 255.1 m | −50% |
| 60 m/s | 38.5° | 154.5 m | 367.3 m | −57.9% |
| 80 m/s | 36.3° | 200.7 m | 653.1 m | −69.3% |
Measured 9 September 2026, K = 0.006, g = 9.8 m/s². The best angle only slides a few degrees, but the range it reaches falls off a cliff as speed rises — the price of pushing a fast object through air. At the 30 m/s optimum the flight is lopsided too: 1.81 s up, 1.95 s down.
The vacuum figures are one line each — for launch speed v at angle θ on level ground:
R = v²·sin(2θ) / g · H = (v·sinθ)² / (2g) · T = 2·v·sinθ / g
The drag figures come from integrating a = −K|v|·v − g with RK4 at a 1/240-second step until the ball lands — the exact routine in the projectile-motion simulator’s solver. Turn drag on there, sweep the angle, and the peak lands where the table says.
Try it with drag on
Open the simulator, switch on air resistance, and sweep the angle — watch the arc shorten, lean, and peak below 45°.