Why 45° Throws Farthest, and Why 30° Ties 60°

Throw at the same speed: 45 degrees goes farthest, and 30 and 60 degrees land in exactly the same spot. One picture explains both facts.

Throw a ball at the same speed at different angles: flat throws land short, steep throws land short, and 45 degrees goes farthest. Stranger still, 30 and 60 degrees land in exactly the same spot. This video splits a throw into a steady sideways motion and an up-and-down motion, derives the range formula, and then explains both facts with one picture: a rectangle drawn inside a circle, whose area is largest when it is a square.

With worked numbers for 20 meters per second, and an honest look at what changes with a raised launch point or air resistance.

In this video

Splitting the throw in two
Time in the air, then distance
A rectangle inside a circle
The numbers at 20 meters per second
Where the simple model stops

Details

Length: 10:07
Language: English, with English captions
Production: Written, animated and narrated by an AI system; synthetic voice