Question

A radioisotope X has a half-life of $10s$ . Find the number of nuclei in the sample (if initially there are 1000 isotopes which are following from rest from a height of 3000m) when it is at a height of 1000m from the reference plane.
(A) 50
(B) 250
(C) 290
(D) 100

Answer 1

Due to free fall, acceleration acting on the body $a$ is acceleration due to gravity $g$ . Initial velocity is zero, since the object falls from rest. Half-life is the time required for a quantity to reduce to half of its initial value.

Formula Used: The formulae used in the solution are given here.
$S = ut + \dfrac{1}{2}a{t^2}$ where $S$ is the distance, $u$ is the initial velocity, $a$ is the acceleration and $t$ is the time taken.
$N\left( t \right) = {N_0}{e^{ - \lambda t}}$ where ${N_0}$ is the initial quantity, $N\left( t \right)$ is the quantity after time $t$ , and $\lambda$ is the decay constant.

Complete Step by Step Solution
It has been given that a radioisotope X has a half-life of $10s$ . Find the number of nuclei in the sample (if initially there are 1000 isotopes which are following from rest from a height of 3000m) when it is at a height of 1000m from the reference plane.
By Newton’s law of motion, we already know that, $S = ut + \dfrac{1}{2}a{t^2}$ .
The distance covered is equal to the height. Thus, height $= S = 2000m.$
Due to free fall, acceleration acting on the body $a$ is acceleration due to gravity $g$ .
Thus, $a = g.$
Initial velocity is zero, since the object falls from rest.
Thus, $S = \dfrac{1}{2}a{t^2}$ .
Now, $a = g$ ,
$2000 = \dfrac{1}{2} \times 10 \times {t^2}$
$\Rightarrow t = \sqrt {\dfrac{{2000}}{5}} = 20s$
Thus time taken in falling a height is 20 seconds.
Given that the half-life is 10 seconds. Half-life is the time required for a quantity to reduce to half of its initial value.
${t_{1/2}} = 10s$ .
We know that, $\lambda = \dfrac{{\ln 2}}{{10}}$ .
Again, $N\left( t \right) = {N_0}{e^{ - \lambda t}}$ where ${N_0}$ is the initial quantity, $N\left( t \right)$ is the quantity after time $t$ , and $\lambda$ is the decay constant.
At $t = 20s$ ,
$N = \dfrac{{{N_0}}}{4} = 250$ .
Thus, the number of nuclei in the sample is $250$ .
The correct answer is Option B.

Note
It has been given that, height ${\text{h = 3000 - 1000 = 2000m}}$
Time taken in falling a height is given as ​​ $t = \sqrt {\dfrac{{2h}}{g}}$ .
Assigning the values, $g = 10m{s^{ - 2}}$ , $h = 2000m$ ,
$t = \sqrt {\dfrac{{2 \times 2000}}{{10}}} = 20s.$
Thus, time taken is 20 seconds.
Number of half-life in this time period is $n = \dfrac{{20}}{{10}} = 2$ .
So the number of active nuclei= ${{initial} \mathord{\left/ {\vphantom {{initial} {{2^n} = {{initial} \mathord{\left/ {\vphantom {{initial} {{2^2}}}} \right. } {{2^2}}}}}} \right. } {{2^n} = {{initial} \mathord{\left/ {\vphantom {{initial} {{2^2}}}} \right. } {{2^2}}}}}$
$= {{initial} \mathord{\left/ {\vphantom {{initial} {4.}}} \right. } {4.}}$
Initially there are 1000 isotopes. The number of active nuclei are ${{1000} \mathord{\left/ {\vphantom {{1000} 4}} \right. } 4} = 250$ .
Option B is correct.