Question

Balls are dropped from the roof of a tower at a fixed interval of time. At the moment when the \[{9^{th}}\] ball reaches the ground \[{n^{th}}\] ball is \[3/{4^{th}}\] height of the tower. The value of n is
A. 13
B. 7
C. 6
D. 5

Answer 1

Hint: Free fall is when an object is dropped from above the earth’s surface and it falls under the force of gravity. This motion will have the effect of acceleration due to gravity. Free fall does not depend on the mass of the falling object and only depends on the height and time taken for the fall.

Formula used :
The height of the object from which it falls,
\[h = \dfrac{1}{2}g{t^2}\]
Where, g – acceleration due to gravity and t – time taken for the free fall.

Complete step by step solution:
When an object moves only with the force due to gravity, then the object is said to be in free fall. Since, the object is acted upon by acceleration due to gravity, the motion will be downwards. During free fall the initial velocity is always zero.

In this case, 9 balls are dropped from the tower at fixed time interval, so the time taken for a ball to fall down is,
\[t = \sqrt {\dfrac{{2h}}{g}} \]
When the \[{9^{th}}\] ball reaches the ground, the \[{n^{th}}\] ball reaches \[3/{4^{th}}\] height of the tower. So, only 8 balls have completely dropped to the ground.

So, the time interval for each ball is,
\[\dfrac{t}{8} = \dfrac{1}{8}\sqrt {\dfrac{{2h}}{g}} \]
Time taken for the \[{n^{th}}\] ball is,
\[{t_n} = (n - 1)\dfrac{1}{8}\sqrt {\dfrac{{2h}}{g}} \]
For, the \[{n^{th}}\] ball the height is \[\dfrac{{3h}}{4}\], from the top the height can be written as \[\dfrac{h}{4}\].
So,
\[\dfrac{h}{4} = \dfrac{1}{2}g{t^2}\]

Substituting t in above equation,
\[\dfrac{h}{4} = \dfrac{1}{2}g{(n - 1)^2}\dfrac{1}{{{{(8)}^2}}}\left( {\dfrac{{2h}}{g}} \right) \\ \]
\[\Rightarrow \dfrac{1}{4} = {(n - 1)^2}\dfrac{1}{{{{(8)}^2}}} \\ \]
\[\Rightarrow \dfrac{{8{\rm{x8}}}}{4} = {(n - 1)^2} \\ \]
\[\Rightarrow {n^2} - 2n + 1 = 16 \]
\[\Rightarrow {n^2} - 2n - 15 = 0\]
Solving the quadratic equation, we get two roots as
\[n = 5,n = - 3\]
Since the number of balls cannot be negative, n = 5 is the answer.

So, the correct answer is option D.

Note : A state of free fall known as weightlessness occurs when the inertial force created by orbital flight or other scenarios that negate gravity completely. It is felt since there is no longer a sense of weight. When no forces from touch are occurring on the bodies, it occurs (human body).