A body of mas m starts from rest with a constant power. If velocity of the body at displacement s is v, then the correct alternative is

Options

A. `s prop v`
B. `s propv^(2)`
C. `s propv^(3)`
D. `sprop v^(4)`

Answer

To solve the problem, we need to establish the relationship between the displacement \( s \) and the final velocity \( v \) of a body of mass \( m \) that starts from rest and moves with constant power \( P \).

### Step-by-Step Solution:

1. **Identify Given Data**:
- Mass of the body: \( m \)
- Initial velocity: \( u = 0 \) (starts from rest)
- Displacement: \( s \)
- Final velocity: \( v \)
- Power: \( P \) (constant)

2. **Use the Power Formula**:
The formula for power \( P \) is given by:
\[ P = F \cdot v \]
where \( F \) is the force acting on the body and \( v \) is its velocity.

3. **Express Force in Terms of Acceleration**:
From Newton's second law, we know:
\[ F = m \cdot a \]
where \( a \) is the acceleration of the body.

4. **Relate Acceleration to Velocity and Displacement**:
Using the third equation of motion:
\[ v^2 = u^2 + 2as \]
Since \( u = 0 \), this simplifies to:
\[ v^2 = 2as \quad \Rightarrow \quad a = \frac{v^2}{2s} \]

5. **Substitute Acceleration into the Force Equation**:
Now substituting \( a \) into the force equation:
\[ F = m \cdot a = m \cdot \frac{v^2}{2s} = \frac{mv^2}{2s} \]

6. **Substitute Force into the Power Equation**:
Now substituting \( F \) back into the power equation:
\[ P = F \cdot v = \left(\frac{mv^2}{2s}\right) \cdot v = \frac{mv^3}{2s} \]

7. **Rearranging for Displacement**:
Rearranging the equation gives:
\[ P \cdot 2s = mv^3 \quad \Rightarrow \quad s = \frac{mv^3}{2P} \]

8. **Establishing the Relationship**:
From the equation \( s = \frac{mv^3}{2P} \), we can see that:
\[ s \propto v^3 \]
This means that displacement \( s \) is directly proportional to the cube of the velocity \( v \).

9. **Conclusion**:
The final relationship we derived is:
\[ s \propto v^3 \]
Therefore, we can conclude that the correct alternative is the one that states \( s \) is proportional to \( v^3 \).