A particle of mass and charge is released from rest in a uniform electric field. If there is no other force on the particle, the dependence of its kinetic energy on the distance travelled by it is correctly given by (graphs are schematic and not drawn to scale)

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the motion of a charged particle in a uniform electric field and determine how its kinetic energy depends on the distance traveled.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Particle:**
- The particle has a mass \( m \) and charge \( q \).
- It is placed in a uniform electric field \( E \).
- The force acting on the particle due to the electric field is given by:
\[ F = qE \]

2. **Apply Newton's Second Law:**
- According to Newton's second law, the net force acting on the particle is equal to the mass of the particle multiplied by its acceleration \( a \):
\[ F = ma \]
- Therefore, we can equate the two expressions for force:
\[ ma = qE \]
- From this, we can express the acceleration \( a \):
\[ a = \frac{qE}{m} \]

3. **Use Kinematic Equations to Find Velocity:**
- Since the particle starts from rest, its initial velocity \( u = 0 \).
- We can use the kinematic equation that relates velocity, initial velocity, acceleration, and displacement:
\[ v^2 = u^2 + 2as \]
- Substituting \( u = 0 \) and \( a = \frac{qE}{m} \), we have:
\[ v^2 = 0 + 2 \left(\frac{qE}{m}\right) x \]
- This simplifies to:
\[ v^2 = \frac{2qE}{m} x \]

4. **Calculate the Kinetic Energy:**
- The kinetic energy \( K \) of the particle is given by:
\[ K = \frac{1}{2} mv^2 \]
- Substituting the expression for \( v^2 \):
\[ K = \frac{1}{2} m \left(\frac{2qE}{m} x\right) \]
- Simplifying this, we find:
\[ K = qEx \]

5. **Determine the Relationship Between Kinetic Energy and Distance:**
- The expression \( K = qEx \) shows that the kinetic energy \( K \) is directly proportional to the distance \( x \) traveled by the particle.
- This indicates a linear relationship, meaning that as the distance increases, the kinetic energy increases linearly.

### Conclusion:
The dependence of the kinetic energy on the distance traveled by the particle is linear, and the correct option is that kinetic energy \( K \) is proportional to \( x \).