Throws & trajectories

Throw a bunch of keys across the room and it does not fly in a straight line but in an arc: first up, then down. The water jet from a garden hose draws the same arc, and so does a basketball on its way to the hoop or a long jumper above the sand pit. This curve is no accident, it follows the same rule every time. Physicists call such a movement projectile motion, and here is the beautiful part: it looks complicated, but it is made of two motions you already know.

Two motions in one

The trick is to split the flight path into two directions. Horizontally, no force acts as long as we ignore air resistance, so the ball simply keeps its pace: a ball that starts forward at 5 m/s is still moving forward at 5 m/s one second later, having covered 5 m. Vertically, gravity pulls, without pause and always downwards: in that direction the ball is in free fall and gains about 9.8 m/s of downward speed every second. The astonishing part is that the two directions do not disturb each other at all. Galileo Galilei was the first to say this clearly, and ever since, one curved path has turned into two calculations you already know how to do.

Falling stays falling

Place two coins at the edge of a table. Let one simply drop and flick the other one away horizontally at the same moment. Which one is down first? Both together, your ears hear only a single click. Vertically the same thing happens to both: free fall from the same height. How long that fall takes depends on the height alone, and from an edge 1 m up it is about 0.45 s. So a harder flick sends the coin farther, but it does not land a fraction of a second later. That makes the range easy: range = horizontal speed times fall time.

Why a parabola?

Put both parts together and you get the arc shape mathematicians call a parabola: steady forward, ever faster downward. If you throw upwards at an angle, gravity first takes vertical speed away from the ball second by second. At the highest point the vertical speed is zero for an instant, while the ball keeps flying forward unhindered, so it never stands still. Then the game reverses and it falls faster and faster while still moving forward. Ignoring air resistance, a ball goes farthest when you release it at 45 degrees. In reality the air slows it down and the throwing hand already starts a metre above the ground, so the best angle in sport is usually a little flatter.