The half life of a particle of mass `1.6 xx 10^(-26) kg` is `6.9 s` and a stream of such particles is travelling with the kinetic energy of a particle being 0.05 eV. The fraction of particles which will decay when they travel a distance of 1 m is

Options

A. `0.1`
B. `0.01`
C. `0.001`
D. `0.0001`

Explanation

To solve the problem, we need to find the fraction of particles that decay when they travel a distance of 1 meter. We will follow these steps:

### Step 1: Calculate the velocity of the particles
The kinetic energy (KE) of a particle is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \(m\) is the mass of the particle and \(v\) is its velocity. We are given:
- Mass \(m = 1.6 \times 10^{-26} \, \text{kg}\)
- Kinetic energy \(KE = 0.05 \, \text{eV}\)

First, we need to convert the kinetic energy from electron volts to joules:
\[ 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \]
Thus,
\[ KE = 0.05 \, \text{eV} = 0.05 \times 1.6 \times 10^{-19} \, \text{J} = 8 \times 10^{-21} \, \text{J} \]

Now, substituting the values into the kinetic energy formula:
\[ 8 \times 10^{-21} = \frac{1}{2} (1.6 \times 10^{-26}) v^2 \]
Rearranging to solve for \(v^2\):
\[ v^2 = \frac{2 \times 8 \times 10^{-21}}{1.6 \times 10^{-26}} = \frac{16 \times 10^{-21}}{1.6 \times 10^{-26}} = 10^{5} \, \text{m}^2/\text{s}^2 \]
Taking the square root gives:
\[ v = 10^{3} \, \text{m/s} \]

### Step 2: Calculate the time taken to travel 1 meter
Using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Velocity}} \]
Substituting the values:
\[ t = \frac{1 \, \text{m}}{10^{3} \, \text{m/s}} = 10^{-3} \, \text{s} \]

### Step 3: Calculate the number of half-lives in that time
The half-life of the particle is given as \(T_{1/2} = 6.9 \, \text{s}\). The number of half-lives \(n\) that fit into the time \(t\) is given by:
\[ n = \frac{t}{T_{1/2}} = \frac{10^{-3}}{6.9} \approx 0.000145 \]

### Step 4: Calculate the fraction of particles that decay
The fraction of particles that remain undecayed after \(n\) half-lives is given by:
\[ \text{Fraction undecayed} = \left(\frac{1}{2}\right)^n \]
Thus, the fraction of particles that decay is:
\[ \text{Fraction decayed} = 1 - \left(\frac{1}{2}\right)^n \]
Substituting \(n \approx 0.000145\):
\[ \text{Fraction undecayed} \approx \left(\frac{1}{2}\right)^{0.000145} \approx 1 - 0.000145 \approx 0.000145 \]

### Final Answer
The fraction of particles that will decay when they travel a distance of 1 meter is approximately:
\[ \text{Fraction decayed} \approx 0.000145 \]