A body moves from rest with a constant acceleration. Which one of the following graphs represents the variation of its kinetic energy K with the distance travelled x?

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining which graph represents the variation of kinetic energy (K) with the distance traveled (x) for a body moving from rest with constant acceleration, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between Kinetic Energy and Distance**:
- The kinetic energy (K) of an object is given by the formula:
\[ K = \frac{1}{2}mv^2 \]
- Where \( m \) is the mass of the body and \( v \) is its velocity.

2. **Relate Velocity to Distance**:
- For an object moving with constant acceleration \( a \) from rest, the velocity \( v \) at any distance \( x \) can be expressed using the kinematic equation:
\[ v^2 = u^2 + 2ax \]
- Since the object starts from rest, \( u = 0 \), so the equation simplifies to:
\[ v^2 = 2ax \]

3. **Substitute Velocity into the Kinetic Energy Formula**:
- Now, substituting \( v^2 \) into the kinetic energy formula:
\[ K = \frac{1}{2}m(2ax) = max \]
- This shows that the kinetic energy \( K \) is directly proportional to the distance \( x \) traveled.

4. **Graphical Representation**:
- Since \( K \) is directly proportional to \( x \), the graph of \( K \) versus \( x \) will be a straight line that passes through the origin (0,0).
- This means that as the distance \( x \) increases, the kinetic energy \( K \) increases linearly.

5. **Identify the Correct Graph**:
- Among the given options, we need to identify the graph that represents a straight line starting from the origin.
- Based on the analysis, the correct graph is the one that shows a linear relationship between kinetic energy and distance, which is represented by option C.

### Conclusion:
The graph that represents the variation of kinetic energy \( K \) with the distance traveled \( x \) is a straight line starting from the origin, corresponding to option C.

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