Newton's Cannon

Placing balls by hand gets old fast. Let's build a cannon.

Launch speed

Fire sideways from the mountain top, and the speed alone picks the path. Slow shots land. At 10, the circular speed, the ball goes round forever. Faster shots swing out wide and come back. Fast enough, and the ball leaves for good:

The paths have names. A circle, an ellipse, a parabola and a hyperbola are all members of one family, and the next lesson draws them with one formula. Here is where each starts:

below 9.8343     an ellipse that dips into the ground: it lands
10               a circle
up to 14.142     an ellipse that comes back
14.142           a parabola: it escapes, just barely
above 14.142     a hyperbola: it escapes with speed to spare

The ball has its own radius, so it touches down a little above the surface. That nudges the landing line up to 9.83.