A body is moving unidirectionally under the influence of a source of constant power supplying energy. Which of the diagrams shown in figure. Correctly shows the displacement-time curve for its motion ?

Options

A.
B.
C.
D.

Explanation

To determine the correct displacement-time curve for a body moving under the influence of a constant power source, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Power**:
- Power (P) is defined as the rate of doing work or the rate of energy transfer. Mathematically, it can be expressed as:
\[ P = \frac{W}{t} = \frac{F \cdot d}{t} \]
where \( W \) is work done, \( F \) is force, \( d \) is displacement, and \( t \) is time.

2. **Relating Power to Force and Velocity**:
- We can also express power in terms of force and velocity:
\[ P = F \cdot v \]
where \( v \) is the velocity of the body. Since power is constant, \( F \cdot v = \text{constant} \).

3. **Using the Relationship of Force and Acceleration**:
- According to Newton's second law, \( F = m \cdot a \), where \( m \) is the mass and \( a \) is acceleration. Since mass is constant, we can write:
\[ P = m \cdot a \cdot v \]

4. **Expressing Acceleration in Terms of Displacement**:
- From the relationship \( a = \frac{dv}{dt} \) and \( v = \frac{ds}{dt} \), we can express acceleration as:
\[ a = \frac{d^2s}{dt^2} \]

5. **Deriving the Displacement-Time Relationship**:
- Since power is constant, we can derive a relationship between displacement and time. By substituting \( a \) into the power equation, we find:
\[ P = m \cdot \frac{d^2s}{dt^2} \cdot \frac{ds}{dt} \]
- Rearranging gives us:
\[ \frac{d^2s}{dt^2} \cdot \frac{ds}{dt} = \frac{P}{m} \]

6. **Integrating the Equation**:
- Integrating this equation leads to a relationship that shows displacement \( s \) is proportional to \( t^{3/2} \):
\[ s \propto t^{3/2} \]

7. **Identifying the Correct Graph**:
- The relationship \( s \propto t^{3/2} \) indicates that as time increases, displacement increases at a rate proportional to \( t^{3/2} \). This means the graph will be a curve that starts at the origin and rises more steeply as time progresses.
- Among the provided options, we need to identify the graph that reflects this relationship.

### Conclusion:
The correct displacement-time curve for the motion of the body under constant power is the one that resembles the function \( s \propto t^{3/2} \), which is a curve that increases more steeply as time increases.