Horizontal Launch: Two Independent Directions

The question

A ball rolls off a table at the same moment another is dropped from the same height. Which lands first?

Both land at the same moment, 0.40 s after release from 0.80 m. Horizontal and vertical motion are independent: gravity changes only the vertical velocity, and both balls start with no vertical velocity from the same height, so they fall together. The sideways speed decides where the rolling ball lands — 1.2 m from the table at 3.0 m/s — not when.

In words, then in symbols

• The distance from the table is the sideways speed times the time it takes to fall the height of the table.

  $x=u\sqrt{{\displaystyle \frac{2h}{g}}}$

How to solve a horizontal launch

1. Split the motion into horizontal and vertical.
2. Vertical: a fall from rest, so use $s=\frac{1}{2}g{t}^2$ to find the time.
3. Horizontal: the velocity never changes, so distance = horizontal velocity × that same time.