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  4. A juggler keeps n balls going with one hand, so that at any instant, (n - 1) balls are in air and one ball in the hand. If each ball rises to a height of x metres, the time for each ball to stay in his hand is

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Q. A juggler keeps $n$ balls going with one hand, so that at any instant, $ (n - 1) $ balls are in air and one ball in the hand. If each ball rises to a height of $ x $ metres, the time for each ball to stay in his hand is

Motion in a Straight Line

Solution:

Let $ u $ be the initial velocity of the ball while going upwards. The final velocity of the ball at height $ x $ is, $ v = 0 $ .
So $ u=\sqrt{2gx} $
Time of flight, $ T=\frac{2u}{g}=\frac{2}{g}\sqrt{2gx}=2 \sqrt{\frac{2x}{g}} $
During time $ T $ , $ (n - 1) $ balls will be in air and one ball will be in hand. So time for one ball in hand
$ =\frac{T}{\left(n-1\right)}=\frac{2\sqrt{2x/g}}{\left(n-1\right)}=\frac{2}{n-1}\sqrt{\frac{2x}{g}} $