Stone · Estonian PhO 2005 #1

Physics · mechanics

Problem

A boy is in the basket of a balloon rising upwards at a constant speed of

v=4\SI{4}{m/s}

. At a certain point, the boy lets a stone fall from the gondola, and then he throws a ball at the stone to try and hit it. What is the longest time the boy can wait between throwing the stone and throwing the ball? Standing on the ground, the boy could throw the ball vertically up to a height of

h = \SI{20}{m}

. It can be assumed that the balloon is high enough for the ball to hit the stone before it reaches the ground. Air resistance is negligible. Take the gravitational acceleration to be

g=\SI{9.8}{m/s^2}

Intuitive explanation

The key insight is that both the stone and ball start with the balloon's upward velocity, so they follow parabolic paths in the ground frame—but what matters is their relative motion. The boy can wait longest when he throws the ball with maximum upward speed (using his full throwing ability of $h = 20\text{ m}$ ), which gives the ball the greatest relative velocity to catch the stone. Find how long the ball can stay in the air after being thrown, then subtract the time the stone needs to reach its maximum height and begin falling back down to where the ball can intercept it.

Intuitive explanation

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