Question

A juggler sets $4$ balls in motion in succession to a height of $19.6m$ from his hand. What is the velocity of projection of each ball? Once set in motion, what are the displacements of the other three balls when the $2nd$ ball just leaves his hand?

Answer 1

$H_{max}=19.6m$

$v^2-u^2=2as$

At maximum height, $v=0$

$u^2=2as$

$u=\sqrt{2g{H}_{max}}$

$u=\sqrt{2\times 9.8\times 19.6}$

$\left|\overrightarrow{u}\right|=19.6m/s\overrightarrow{u}=19.6m/s\hat{j}$

Time of flight of each ball $=\frac{2u}{g}=\frac{2\times 19.6}{9.8}=4s$

We can see,juggler has to manage 4 balls each having time of flight 4s

He has to manage the time in this symmetrically way to get his job done.

$S_1$ =displacement of first ball

$S_2$ =displacement of third ball

$S_1$ =displacement of fourth ball

$S_1=ut-\frac{1}{2}g{t}^2$

$=19.6\times 1-\frac{1}{2}\times 9.8\times 1=14.7m$

$\overrightarrow{S_1}=14.7\hat{j}\left(m\right)$

$\overrightarrow{S_3}=19.6\times 2-\frac{1}{2}\times 9.8\times 4$

$\overrightarrow{S_3}=19.6\hat{j}\left(m\right)$

$\overrightarrow{S_4}=19.6\times 3-\frac{1}{2}\times 9.8\times 9$

$\overrightarrow{S_4}=14.7\hat{j}\left(m\right)$