An object is dragged along a horizontal surface by the machine which delivers a constant power. How does the distance moved by the object is dependent on time, t ?

Options

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

Explanation

To determine how the distance moved by an object, \( s \), is dependent on time, \( t \), when the object is dragged by a machine delivering constant power, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Power**:
The power \( P \) delivered by the machine is constant. Power is defined as the product of force \( F \) and velocity \( v \):
\[ P = F \cdot v \]

2. **Relating Force and Acceleration**:
According to Newton's second law, force can also be expressed as:
\[ F = m \cdot a \]
where \( m \) is the mass of the object and \( a \) is its acceleration.

3. **Expressing Velocity**:
The velocity \( v \) of the object can be expressed in terms of acceleration and time:
\[ v = u + at \]
Given that the initial velocity \( u = 0 \) (the object starts from rest), this simplifies to:
\[ v = at \]

4. **Substituting into Power Equation**:
Substituting \( F = m \cdot a \) and \( v = at \) into the power equation gives:
\[ P = m \cdot a \cdot (at) = m \cdot a^2 t \]

5. **Solving for Acceleration**:
Rearranging the equation to solve for acceleration \( a \):
\[ a = \sqrt{\frac{P}{m t}} \]

6. **Using the Kinematic Equation**:
The distance \( s \) traveled by the object can be calculated using the equation:
\[ s = ut + \frac{1}{2} a t^2 \]
Since \( u = 0 \), this simplifies to:
\[ s = \frac{1}{2} a t^2 \]

7. **Substituting for Acceleration**:
Substituting the expression for \( a \) into the distance equation:
\[ s = \frac{1}{2} \left( \sqrt{\frac{P}{m t}} \right) t^2 \]
This can be simplified to:
\[ s = \frac{1}{2} \sqrt{\frac{P}{m}} t^{3/2} \]

8. **Final Relationship**:
Thus, we find that the distance \( s \) is proportional to \( t^{3/2} \):
\[ s \propto t^{3/2} \]

### Conclusion:
The distance moved by the object is dependent on time \( t \) as:
\[ s \propto t^{3/2} \]