Answer to Question #114077 in Classical Mechanics for Loriel Smith

Question

If a ball is falling vertically downwards and another ball is rolling down from an inclined plane (friction and air drag are absent)at the same time and from same height. Then which ball will reach to the ground first and why? (ASSUMING initial velocities of both balls are zero)

Answer 1

According to the law of conservation of energy, both of them will reach the bottom at the same velocity if started at the same height h:

$$mgh=\frac{1}{2}mv^2,\\ v=\sqrt{2gh}.$$

However, since the ball rolling on the incline will have the longer way to reach the ground, it will reach the bottom after the ball that was dropped:

$$t=\sqrt{\frac{2h}{a}},\\ a=\frac{v^2}{2s}=\frac{v^2\text{ sin}\theta}{2h}=g\text{ sin}\theta,\\ t=\sqrt{\frac{2h}{g\text{ sin}\theta}}.$$

where theta - the angle of the incline above the horizontal.

The dropped ball will reach the ground faster:

$$\tau=\sqrt{\frac{2h}{g}}<t.$$