A particle of mass `1 mg` has the same wavelength as an electron moving with a velocity of `3 xx 10^(6) ms^(-1)`. The velocity of the particle is

Options

A. `2.7 xx 10^(-18)ms^(-1)`
B. `9 xx 10^(-2) ms_(1)`
C. `3 xx 10^(-31)ms_(-1)`
D. `2.7 xx 10^(-21) ms^(-1)`

Answer

To solve the problem, we need to find the velocity of a particle of mass \(1 \, \text{mg}\) that has the same wavelength as an electron moving with a velocity of \(3 \times 10^6 \, \text{m/s}\).

### Step-by-Step Solution:

1. **Understand the De Broglie Wavelength Formula**:
The de Broglie wavelength (\(\lambda\)) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \(h\) is Planck's constant and \(p\) is the momentum of the particle.

2. **Set Up the Equation**:
Since the particle and the electron have the same wavelength, we can equate their wavelengths:
\[ \lambda_p = \lambda_e \]
This leads to:
\[ \frac{h}{p_p} = \frac{h}{p_e} \]
which simplifies to:
\[ p_p = p_e \]

3. **Express Momentum**:
The momentum \(p\) of a particle is given by:
\[ p = mv \]
Therefore, for the particle and the electron:
\[ m_p v_p = m_e v_e \]
where:
- \(m_p\) = mass of the particle
- \(v_p\) = velocity of the particle
- \(m_e\) = mass of the electron (\(9.1 \times 10^{-31} \, \text{kg}\))
- \(v_e\) = velocity of the electron (\(3 \times 10^6 \, \text{m/s}\))

4. **Substitute Known Values**:
We know:
- Mass of the particle \(m_p = 1 \, \text{mg} = 1 \times 10^{-3} \, \text{g} = 1 \times 10^{-6} \, \text{kg}\)
- Mass of the electron \(m_e = 9.1 \times 10^{-31} \, \text{kg}\)
- Velocity of the electron \(v_e = 3 \times 10^6 \, \text{m/s}\)

5. **Rearranging for Velocity of the Particle**:
From the momentum equation:
\[ v_p = \frac{m_e v_e}{m_p} \]

6. **Substituting the Values**:
\[ v_p = \frac{(9.1 \times 10^{-31} \, \text{kg}) \cdot (3 \times 10^6 \, \text{m/s})}{1 \times 10^{-6} \, \text{kg}} \]

7. **Calculating**:
\[ v_p = \frac{(9.1 \times 3) \times 10^{-25}}{1 \times 10^{-6}} = \frac{27.3 \times 10^{-25}}{1 \times 10^{-6}} = 27.3 \times 10^{-19} \, \text{m/s} \]

8. **Final Result**:
\[ v_p = 2.73 \times 10^{-18} \, \text{m/s} \]

### Final Answer:
The velocity of the particle is \(2.73 \times 10^{-18} \, \text{m/s}\).