If the material particle is moving with velocity v and the velocity of light is c, then mass of particle is taken as (`m_0` is rest mass)

Options

A. `m=m_0/c`
B. `m=m_0/(√(1- v^2/c^2))`
C. `m=m_0√(1+v^2/c^2)`
D. `m=m_0/(√(1+v^2/c^2))`

Answer

To solve the problem regarding the mass of a particle moving with velocity \( v \) in relation to its rest mass \( m_0 \), we can use the concepts of relativistic physics, particularly the Lorentz factor \( \gamma \).

### Step-by-step Solution:

1. **Understanding the Lorentz Factor**:
The Lorentz factor \( \gamma \) is defined as:
\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]
where \( v \) is the velocity of the particle and \( c \) is the speed of light.

2. **Relating Moving Mass to Rest Mass**:
The relativistic mass \( m \) of a particle moving at velocity \( v \) is related to its rest mass \( m_0 \) by the equation:
\[ m = \gamma m_0 \]

3. **Substituting the Lorentz Factor**:
By substituting the expression for \( \gamma \) into the equation for moving mass, we get:
\[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]

4. **Final Expression**:
This equation shows how the mass of the particle increases as its velocity approaches the speed of light. Thus, the mass of the particle moving with velocity \( v \) is given by:
\[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]

### Summary:
The mass of a particle moving with velocity \( v \) in relation to its rest mass \( m_0 \) is expressed as:
\[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]