Question

A stone falls freely from rest from a height h and it travels a distance $\frac{9h}{25}$ in the last second. The value of h is (Take $g=10m/s^2$):

• A. $145m$
• B. $100m$
• C. $125m$
• D. $200m$

Answer 1

The correct option is C $125m$

$\text{Step 1: Equations of motion for Last second}$

Let the total time of fall is 't' s, in the last second ($t^{th}$ second) displacement is $\frac{9h}{25}$

Initial speed $u=0$ $;$ Acceleration $a=g$

Using expression for displacement in $n^{th}$ second (Taking downward positive)

$S_n=u+\frac{a}{2}(2n-1)$

$\therefore \frac{9h}{25}=\frac{g}{2}(2t-1)$ $....\left(1\right)$

$\text{Step 2: Equations of motion for whole motion}$

(Taking downward Positive)

$s=ut+\frac{1}{2}a{t}^2$

Putting $s=h$ and $u=0$

$\Rightarrow h=\frac{1}{2}g{t}^2$ ....(2)

$\text{Step 3: Equation solving}$

From equation $\left(1\right)$ and $\left(2\right)$

$\frac{25g}{18}=\frac{1}{2}g{t}^2$

$9t^2-50t+25$

$\Rightarrow t=5s$

From equation $\left(2\right)$

$h=\frac{1}{2}\times 10\times (5{)}^2$

$=125m$

Hence the value of $h$ is $125m$.

Option $C$.