Launch Angle
y = max(x · tan(deg(45) + deg(30) · sin(t)) −((9.8x²)/(2 · v₀² · cos(deg(45) + deg(30) · sin(t))²)), 0)
Open in the app The dials and keys named below are the app's.
What it draws
This is the path of a projectile thrown from the origin, with x the distance downrange and the value the height. Eliminating the time of flight from the two equations of motion leaves the height as x·tan(θ) − g·x²/(2·v₀²·cos²(θ)), a straight rise at the launch angle with a parabolic droop subtracted from it. Gravity is written into the formula as the number 9.8, the value on the Earth, so the one letter left is v₀, the launch speed, opening at 8.
The launch angle is not a letter but the clock. θ = 45° + 30°·sin t swings between 15° and 75° and back once every 2π ≈ 6.28 seconds at Speed 1. The outer max(…, 0) is the ground. Past the landing point the parabola dives steeply, so the floor cuts it off and each frame reads as flat ground, one arc, flat ground again.
Why 45° wins
The parabola returns to zero at x = v₀²·sin(2θ)/g, which is 6.53 at 45° and the longest throw the sweep reaches. Because sin(2θ) is unchanged by swapping θ for 90° − θ, the shallow and the steep throw land together: 15° and 75° both reach 3.27. The apex is v₀²·sin²(θ)/(2g), which is 0.22 for the flat throw and 3.05 for the lobbed one, a factor of 14 between two arcs of identical range.
That symmetry shows in the timing: the angle passes 45° twice in each swing, at t = 0 and t = π, so the landing point runs out and back twice while the arc rises and falls once.
What the launch speed does
The letter changes the size of the picture and not the shapes in it, because the range depends on speed and gravity only through v₀²/g. Halving v₀ shrinks every arc of the sweep to a quarter, and doubling it stretches them fourfold. Only the square of the speed appears, so the sign is immaterial and −8 throws exactly as 8 does. At v₀ = 0 the droop is infinite everywhere, the zero floor catches all of it, and only flat ground is left.
Try
- Drag v₀ to 10: the 45° throw reaches 10.20, because range follows the square of the launch speed, and still lands inside the opening view.
- Drag v₀ to 4: every arc shrinks to a quarter of the one beside it, the longest throw reaching 1.63.
- Change the 9.8 to 3.7, roughly the gravity of Mars: the same throw now carries 17.30, so raise Span of x to 18 to watch it land.
- Change both 30° to 44°: the sweep then runs from 1° to 89°, and both ends of it barely clear the launch point.
- Top in the deck: from above, the ribbon's outer edge traces the landing point against time, touching the near mark twice a swing as the shallow throw and the steep one land together.