A body is moved along a straight line by a machine delivering constant power . The distance moved by the body is time `t` is proptional to

Options

A. (a) `t^(1//2)`
B. (b) `t^(3//4)`
C. (c) `t^(3//2)`
D. (d) `t^2`

Explanation

To solve the problem, we need to analyze the relationship between power, force, velocity, and displacement when a body is moved by a machine delivering constant power.

### Step-by-Step Solution:

1. **Understand the relationship between power, force, and velocity:**
\[ P = F \cdot v \]
where \( P \) is power, \( F \) is force, and \( v \) is velocity.

2. **Express force in terms of mass and acceleration:**
\[ F = m \cdot a \]
where \( m \) is mass and \( a \) is acceleration.

3. **Relate acceleration to velocity and displacement:**
Acceleration can be expressed as:
\[ a = \frac{dv}{dt} = \frac{d^2x}{dt^2} \]
where \( x \) is displacement.

4. **Substitute acceleration into the power equation:**
Replacing \( F \) in the power equation gives:
\[ P = m \cdot a \cdot v \]
Substituting \( a \) with \( \frac{dv}{dt} \) and \( v \) with \( \frac{dx}{dt} \):
\[ P = m \cdot \frac{d^2x}{dt^2} \cdot \frac{dx}{dt} \]

5. **Rearranging the equation:**
This can be rearranged to:
\[ P = m \cdot \frac{dx}{dt} \cdot \frac{d^2x}{dt^2} \]
or
\[ P = m \cdot v \cdot a \]

6. **Express velocity in terms of displacement and time:**
Velocity can be expressed as:
\[ v = \frac{dx}{dt} \]

7. **Substituting velocity into the power equation:**
Thus, we can write:
\[ P = m \cdot \frac{dx}{dt} \cdot \frac{d^2x}{dt^2} = m \cdot \frac{dx}{dt} \cdot \frac{d}{dt}\left(\frac{dx}{dt}\right) \]

8. **Using the relationship between displacement and time:**
We can express \( v \) as \( \frac{dx}{dt} \) and substitute back into the power equation:
\[ P = m \cdot \frac{dx}{dt} \cdot \frac{d^2x}{dt^2} = m \cdot \frac{dx}{dt} \cdot \frac{d}{dt}\left(\frac{dx}{dt}\right) \]

9. **Integrating to find displacement:**
By integrating and rearranging, we find:
\[ P = \frac{m \cdot x^2}{t^3} \]
Rearranging gives:
\[ x^2 = \frac{P \cdot t^3}{m} \]

10. **Taking the square root to find displacement:**
\[ x = \sqrt{\frac{P \cdot t^3}{m}} = \frac{\sqrt{P}}{\sqrt{m}} \cdot t^{3/2} \]

11. **Conclusion:**
Thus, we find that the displacement \( x \) is proportional to \( t^{3/2} \):
\[ x \propto t^{3/2} \]

### Final Answer:
The distance moved by the body in time \( t \) is proportional to \( t^{3/2} \).

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