Question

A juggler throws the balls in the air. How high will each ball rise if he throws n balls per second?
A. \[\dfrac{{4{{\rm{n}}^2}}}{g}\]
B. \[\dfrac{g}{{{\rm{2}}{{\rm{n}}^2}}}\]
C. \[\dfrac{g}{{4{\pi ^2}}}\]
D. \[\dfrac{{4g}}{{{{\rm{n}}^2}}}\]

Answer 1

Hint: We will be using the equations of motion, mainly the first and third equation. These equations tell us about the relation between initial velocity, final velocity, acceleration and time period of a moving body.

Complete step by step answer:
First equation of motion of a body under the influence of gravity.
\[v = u - gt\]……(1)

Here v is initial velocity, u is final velocity and t is time period.

Third equation of motion of a body under the influence of gravity.
\[{v^2} = {u^2} - 2gs\]……(2)
\[\]
Here s is the distance travelled.

When a ball is thrown upward, at the instant of maximum height attained its velocity is zero.
V=0 m/s

Here v is the final velocity of a ball from a number of balls thrown upward by the juggler.

It is given that the juggler throws n number of balls in one second so the time taken to throw one ball will be given as \[\left( {\dfrac{1}{{\rm{n}}}} \right)\] second.

Substitute \[\dfrac{1}{{\rm{n}}}\] for t and \[0\] for v in equation (1).
\[\begin{array}{l}
v = u - g\left( {\dfrac{1}{{\rm{n}}}} \right)\\
u = \dfrac{g}{{\rm{n}}}
\end{array}\]

Let H is the maximum height attained by each ball.

Substitute H for s, \[0\] for v and \[\dfrac{g}{{\rm{n}}}\] for u in equation (2).
\[\begin{array}{l}
{0^2} = {\left( {\dfrac{g}{{\rm{n}}}} \right)^2} - 2gH\\
H = \dfrac{g}{{2{{\rm{n}}^2}}}
\end{array}\]

Therefore, maximum height attained by each ball if juggler throws n balls per second is given by \[\left( {\dfrac{g}{{2{{\rm{n}}^2}}}} \right)\]

So, the correct answer is “Option B”.

Note:
Take extra care while using the equations of motion because under the influence of gravity their forms have slight changes than conventional ones.