A machine is delivering constant power to drive a body along a straight line. What is the relation between the distance travelled by the body against time ?

Options

A. (a)`s^(2)alpha t^(3)`
B. (b)`s^(2)alpha t^(3)`
C. (c)`s^(3)alpha t^(2)`
D. (d)`s alpha t^(3)`

Answer

To solve the problem of finding the relationship between the distance traveled by a body and time when a machine delivers constant power, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Power Equation**:
We know that power (P) is defined as the product of force (F) and velocity (v):
\[ P = F \cdot v \]

2. **Express Force in Terms of Mass and Acceleration**:
The force can be expressed using Newton's second law:
\[ F = m \cdot a \]
where \( m \) is the mass of the body and \( a \) is its acceleration.

3. **Relate Acceleration to Distance and Time**:
Acceleration can also be expressed as the change in velocity over time. If we consider distance (s) traveled in time (t), we can express acceleration as:
\[ a = \frac{s}{t^2} \]
This is derived from the kinematic equation \( s = \frac{1}{2} a t^2 \) when starting from rest.

4. **Substitute Acceleration into the Power Equation**:
Now, substituting \( F \) and \( v \) into the power equation:
- Velocity \( v \) can be expressed as \( v = \frac{s}{t} \).
- Therefore, substituting \( F \) and \( v \):
\[ P = (m \cdot a) \cdot v = m \cdot \left(\frac{s}{t^2}\right) \cdot \left(\frac{s}{t}\right) \]
Simplifying this gives:
\[ P = m \cdot \frac{s^2}{t^3} \]

5. **Rearranging the Equation**:
Since power (P) is constant, we can rearrange the equation to find the relationship between distance and time:
\[ s^2 = \frac{P \cdot t^3}{m} \]
This shows that \( s^2 \) is directly proportional to \( t^3 \):
\[ s^2 \propto t^3 \]

6. **Conclusion**:
The relationship between the distance traveled (s) and time (t) is:
\[ s^2 = k \cdot t^3 \]
where \( k = \frac{P}{m} \) is a constant.

### Final Relation:
Thus, we conclude that the distance squared is directly proportional to the cube of time:
\[ s^2 \propto t^3 \]