Question

A ball is released from the top of a tower of height $h$ meters. It takes $T$ seconds to reach the ground. What is the position of the ball in $T/3$ second?
A. $2h/9$ meter from the ground
B. $7h/9$ meter from the ground
C. $8h/9$ meter from the ground
D. $17h/18$ meter from the ground

Answer 1

Derive the formula for distance of the ball from top of the tower in terms of time required for the ball to reach ground. Substitute the time $T/3$ in this derivation and determine the position of the ball at time $T/3$ from the top of the tower. Then subtract the obtained value from height of the tower to determine its position from the ground.

Formula used:
The kinematic equation for displacement $s$ of the object is given by
$s = ut + \dfrac{1}{2}a{t^2}$ …… (1)
Here, $u$ is the initial velocity of the object, $a$ is the acceleration of object and $t$ is the time.

Complete step by step answer:
We have given that a ball is released from the top of a tower whose height is $h$. The time required for the ball to travel the distance from top of the tower to the ground is $T$. The acceleration of the ball while moving downward is equal to acceleration due to gravity $g$.
$a = g$

The initial velocity $u$ of the ball at the top of the tower is zero as it starts from the rest.
$u = 0\,{\text{m/s}}$
The displacement of the ball is equal to the height of the tower.
$s = h$

Substitute $h$ for $s$, $0\,{\text{m/s}}$ for $u$, $T$ for $t$ and $g$ for $a$ in equation (1).
$h = \left( {0\,{\text{m/s}}} \right)T + \dfrac{1}{2}g{T^2}$
$\Rightarrow h = \dfrac{1}{2}g{T^2}$ …… (2)
Here, we have asked to determine the position of the ball in $T/3$ second.
Substitute $\dfrac{T}{3}$ for $T$ in equation (2).
${h_1} = \dfrac{1}{2}g{\left( {\dfrac{T}{3}} \right)^2}$
$\Rightarrow {h_1} = \dfrac{1}{2}\dfrac{{g{T^2}}}{9}$
Substitute $h$ for $\dfrac{1}{2}g{T^2}$ in the above equation.
$\Rightarrow {h_1} = \dfrac{h}{9}$
Hence, the distance of the ball from the top of the tower is $\dfrac{h}{9}$.

Now let us determine the position of the ball from the ground. The position $H$ of the ball from the ground is given by
$H = h - \dfrac{h}{9}$
$\therefore H = \dfrac{{8h}}{9}$
Therefore, the position of the ball is $\dfrac{{8h}}{9}$ meter from the ground.

Hence, the correct option is C.

Note: The students should not forget to subtract the distance of the ball obtained from the total height of the tower. If this value is not subtracted from the height of the tower, it will be the distance of the ball from top of the tower and not from the ground which is the required position of the ball if we see the options.