The relation between kinetic energy K and linear momentum p of a particle is represented by

Options

A. `K^(2)= 2mp`
B. `2K^(2)= mp`
C. `p^(2)=2mK`
D. `2mK= p`

Explanation

To find the relation between kinetic energy \( K \) and linear momentum \( p \) of a particle, we can start from the definitions of both quantities.

### Step-by-Step Solution:

1. **Define Kinetic Energy and Momentum**:
- The kinetic energy \( K \) of a particle is given by the formula:
\[ K = \frac{1}{2} mv^2 \]
- The linear momentum \( p \) of a particle is defined as:
\[ p = mv \]

2. **Express Velocity in Terms of Momentum**:
- From the momentum equation, we can express the velocity \( v \) as:
\[ v = \frac{p}{m} \]

3. **Substitute Velocity into Kinetic Energy Formula**:
- Now, substitute \( v \) in the kinetic energy formula:
\[ K = \frac{1}{2} m \left(\frac{p}{m}\right)^2 \]

4. **Simplify the Expression**:
- Simplifying the above expression gives:
\[ K = \frac{1}{2} m \cdot \frac{p^2}{m^2} = \frac{p^2}{2m} \]

5. **Rearranging the Equation**:
- To express the relationship in terms of \( p^2 \), we can rearrange the equation:
\[ p^2 = 2mK \]

### Final Relation:
Thus, the relation between kinetic energy \( K \) and linear momentum \( p \) is:
\[ p^2 = 2mK \]