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

### Step 1: Understand the relationship between power, force, and velocity
Power (P) is defined as the product of force (F) and velocity (V):
\[ P = F \cdot V \]
Since the power is constant, we can express this as:
\[ F = \frac{P}{V} \]

### Step 2: Relate force to mass and acceleration
According to Newton's second law, force is also given by:
\[ F = m \cdot a \]
where \( m \) is the mass of the body and \( a \) is its acceleration. Therefore, we can write:
\[ m \cdot a = \frac{P}{V} \]

### Step 3: Express acceleration in terms of velocity and displacement
Since acceleration \( a \) can be expressed as:
\[ a = \frac{dV}{dt} \]
and using the chain rule:
\[ a = V \frac{dV}{dx} \]
we can substitute this into our equation:
\[ m \cdot V \frac{dV}{dx} = \frac{P}{V} \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ mV^2 \frac{dV}{dx} = P \]
Now, we can separate variables:
\[ V^2 dV = \frac{P}{m} dx \]

### Step 5: Integrate both sides
Integrating both sides:
\[ \int V^2 dV = \int \frac{P}{m} dx \]
This results in:
\[ \frac{V^3}{3} = \frac{P}{m} x + C \]
Assuming the initial conditions where \( V = 0 \) when \( x = 0 \), we find that \( C = 0 \). Thus:
\[ V^3 = \frac{3P}{m} x \]

### Step 6: Express velocity in terms of displacement
Taking the cube root gives us:
\[ V = \left(\frac{3P}{m}\right)^{1/3} x^{1/3} \]

### Step 7: Relate velocity to time
Since velocity \( V \) is also defined as:
\[ V = \frac{dx}{dt} \]
we can set up the equation:
\[ \frac{dx}{dt} = \left(\frac{3P}{m}\right)^{1/3} x^{1/3} \]

### Step 8: Separate variables and integrate again
Separating variables gives:
\[ \frac{dx}{x^{1/3}} = \left(\frac{3P}{m}\right)^{1/3} dt \]
Integrating both sides:
\[ \int x^{-1/3} dx = \left(\frac{3P}{m}\right)^{1/3} \int dt \]
This results in:
\[ \frac{3}{2} x^{2/3} = \left(\frac{3P}{m}\right)^{1/3} t + C \]
Again, using initial conditions, we find \( C = 0 \):
\[ x^{2/3} = \frac{2}{3} \left(\frac{3P}{m}\right)^{1/3} t \]

### Step 9: Solve for displacement
Finally, solving for \( x \):
\[ x = \left(\frac{2P}{m}\right)^{1/2} t^{3/2} \]

### Step 10: Identify the displacement-time relationship
The relationship \( x \propto t^{3/2} \) indicates that the displacement increases with the power of \( t^{3/2} \). This means the displacement-time graph will be a curve that rises more steeply as time increases.

### Conclusion
The correct displacement-time curve will be a curve that represents \( x \) as a function of \( t^{3/2} \), which corresponds to option B in the provided diagrams.

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