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Last updated date: 02nd Dec 2023
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# A juggler throws balls into the air. He throws one whenever the previous one is at its highest point. How high do the halls rise if he throws $n$ balls each second ?

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Hint:When a ball is thrown upwards then there is some initial velocity $u$ and a final velocity $v$. The final velocity becomes zero at the highest point. And the force which works on the ball in this situation is the gravitational force, hence the acceleration a becomes -g(acceleration due to gravity is in opposite direction of motion when ball goes from ground to top).

Given that the juggler throws $n$ balls in one second. It implies that one ball takes $\dfrac{1}{n}$ seconds to reach at the highest point. Three equations of motion are:
First equation: $v~=~u~+at$
Second equation: $s=ut+\dfrac{1}{2}a{{t}^{2}}$
Third equation: $2as={{v}^{2}}-{{u}^{2}}$
Let’s this time be t seconds, i.e. $t=\dfrac{1}{n}$.

When a ball is thrown upwards then there is some initial velocity u and a final velocity $v$.The final velocity becomes zero at the highest point. And the force which works on the ball in this situation is the gravitational force, hence the acceleration a becomes -g(acceleration due to gravity is in opposite direction of motion when ball goes from ground to top).

Now putting $v=0$ and $a=-g$in the first equation of motion, $v=u+at$, we get: $u=\dfrac{g}{n}$. It means the initial velocity with which each ball is thrown upwards is $\dfrac{g}{n}$. Now let’s consider the motion to the highest point; initial velocity $u=\dfrac{g}{n}$, final velocity $v=0$, total distance (s)= maximum height that a ball gains (h). Applying third equation of motion and putting the above quantities, we observe that:
$2as={{v}^{2}}-{{u}^{2}}$ changes to $2gh={{0}^{2}}-\dfrac{{{g}^{2}}}{{{n}^{2}}}$.
$\therefore h=\dfrac{g}{2{{n}^{2}}}$

Therefore the maximum height taken by each ball when a juggler throws n balls in one second and also each ball is thrown whenever the previous one is at its highest point is $\dfrac{g}{2{{n}^{2}}}$.

Note:According to the question $n$ balls are thrown each second, which means $n$ balls are thrown in one second. Then by unitary method we can say that one ball is thrown in $\dfrac{1}{n}$ seconds. You can also directly calculate, without using the unitary method.