A radioiostope `X` has a half-life of `10 s`. Find the number of nuclei in the sample (if initially there are `1000` isotopes which are falling from rest from a height of `3000 m` ) when it is at a height of `1000 m` from refrence plane.

Options

A. `50`
B. 2`50`
C. `29`
D. `100`

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Determine the distance fallen
The initial height from which the isotopes are falling is 3000 m, and we need to find the number of nuclei when they are at a height of 1000 m. Therefore, the distance fallen (s) is:
\[ s = 3000 \, \text{m} - 1000 \, \text{m} = 2000 \, \text{m} \]

**Hint:** Calculate the distance fallen by subtracting the final height from the initial height.

### Step 2: Calculate the time taken to fall 2000 m
Using the equation of motion for free fall:
\[ s = ut + \frac{1}{2} a t^2 \]
where \( u = 0 \) (initial velocity), \( a = g \) (acceleration due to gravity, approximately \( 10 \, \text{m/s}^2 \)), and \( s = 2000 \, \text{m} \):
\[ 2000 = 0 + \frac{1}{2} \cdot 10 \cdot t^2 \]
This simplifies to:
\[ 2000 = 5t^2 \]
Thus,
\[ t^2 = \frac{2000}{5} = 400 \implies t = \sqrt{400} = 20 \, \text{s} \]

**Hint:** Use the equation of motion for objects in free fall to find the time taken to fall a certain distance.

### Step 3: Determine the number of half-lives elapsed
The half-life of the isotope \( X \) is given as \( 10 \, \text{s} \). Since the time taken to fall 2000 m is \( 20 \, \text{s} \), we can find the number of half-lives that have passed:
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{20 \, \text{s}}{10 \, \text{s}} = 2 \]

**Hint:** Divide the total time by the half-life to find how many half-lives have passed.

### Step 4: Calculate the remaining number of nuclei
The initial number of nuclei \( N_0 \) is \( 1000 \). After each half-life, the number of nuclei is halved. After 2 half-lives:
\[ N_t = N_0 \left( \frac{1}{2} \right)^n \]
where \( n \) is the number of half-lives. Thus:
\[ N_t = 1000 \left( \frac{1}{2} \right)^2 = 1000 \cdot \frac{1}{4} = 250 \]

**Hint:** Use the formula for radioactive decay to find the remaining quantity after a certain number of half-lives.

### Final Answer
The number of nuclei remaining in the sample when it is at a height of 1000 m is **250**.

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