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

Options

A. `t^(1//2)`
B. `t^(3//4)`
C. `t^(3//2)`
D. `t^(2)`

Explanation

To solve the problem step by step, we need to analyze the relationship between power, distance, and time when a body is moved by a machine delivering constant power.

### Step-by-Step Solution:

1. **Understanding Power**:
Power (P) is defined as the rate at which work is done or energy is transferred. Mathematically, it is given by:
\[ P = \frac{E}{t} \]
where \(E\) is the energy and \(t\) is the time.

2. **Kinetic Energy Consideration**:
Since the body is moving along a straight line, we can consider its kinetic energy (KE). The kinetic energy of a body with mass \(m\) moving with velocity \(v\) is given by:
\[ KE = \frac{1}{2} mv^2 \]

3. **Relating Power to Kinetic Energy**:
Since the power is constant, we can express it as:
\[ P = \frac{KE}{t} = \frac{\frac{1}{2} mv^2}{t} \]
Rearranging gives us:
\[ mv^2 = 2Pt \]
This implies that \(v^2\) is proportional to \(t\) since \(m\) and \(P\) are constants.

4. **Expressing Velocity**:
From the equation \(mv^2 = 2Pt\), we can express the velocity \(v\) as:
\[ v = \sqrt{\frac{2P}{m}} \sqrt{t} \]
Let \(k = \sqrt{\frac{2P}{m}}\), then:
\[ v = kt^{1/2} \]

5. **Relating Velocity to Distance**:
The velocity \(v\) can also be expressed in terms of distance \(x\) and time \(t\):
\[ v = \frac{dx}{dt} \]
Therefore, we can write:
\[ \frac{dx}{dt} = kt^{1/2} \]

6. **Integrating to Find Distance**:
To find the total distance \(x\) moved in time \(t\), we integrate both sides:
\[ dx = kt^{1/2} dt \]
Integrating gives:
\[ x = \int kt^{1/2} dt = k \cdot \frac{t^{3/2}}{3/2} + C_1 = \frac{2k}{3} t^{3/2} + C_1 \]
where \(C_1\) is the constant of integration.

7. **Proportionality**:
Since we are interested in the proportionality of distance \(x\) with respect to time \(t\), we can ignore the constant \(C_1\) and write:
\[ x \propto t^{3/2} \]

8. **Conclusion**:
Thus, the distance moved by the body in time \(t\) is proportional to \(t^{3/2}\).

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