Why 45 degrees throws the farthest, and when it does not
The physics behind the optimal launch angle, why raising the release point lowers it, and where drag breaks the tidy rule.
The symmetric ground-level case
When a projectile launches and lands at the same height, its flight is perfectly symmetric: the time going up equals the time coming down. The horizontal range then depends on the product of horizontal speed and total time, and both come from splitting the launch speed by angle. Balancing more horizontal speed against more air time is what makes a middle angle win. That balance point is exactly 45 degrees on level ground.
Why the sine of twice the angle appears
The range formula carries a sin(2 x angle) term, and a sine reaches its maximum at 90 degrees. Since the angle inside is doubled, the peak lands at a launch angle of 45 degrees. This also explains a neat symmetry: a 30 degree and a 60 degree launch at the same speed cover the same distance, because their doubled angles, 60 and 120, share the same sine. The calculator reproduces this pairing exactly.
Raising the launch point
Throwing from a height breaks the up-down symmetry, because the object falls farther than it rose. That bonus fall time rewards keeping more of the launch speed horizontal, so the range-maximising angle drops below 45 degrees. The higher and faster the launch, the more the optimum shifts toward flatter throws. You can see the effect by setting a launch height and comparing the range at 45 degrees against a slightly lower angle.
Where real air spoils the ideal
These clean rules assume no air resistance, so they suit dense, slow objects like a thrown rock or a physics-class ball. Real drag steals energy, shortens the range and pulls the best angle well below 45 degrees for fast or light objects. A driven golf ball or a thrown javelin is optimised nearer 35 to 40 degrees once aerodynamics are included. Treat this tool as the frictionless baseline that real trajectories fall short of.