A constant force F acts on a body The power delivered by F will depend on position x as (velocity of body is zero at x = 0)

Options

A. `X^(2/3)`
B. `X^(1/3)`
C. `X^(3/5)`
D. `X^(1/2)`

Answer

To solve the problem, we need to determine how the power delivered by a constant force \( F \) acting on a body depends on the position \( x \), given that the initial velocity of the body is zero at \( x = 0 \).

### Step-by-Step Solution:

1. **Understanding Power**:
The power \( P \) delivered by a force \( F \) is given by the formula:
\[ P = F \cdot v \]
where \( v \) is the velocity of the body.

2. **Using the Equation of Motion**:
Since the initial velocity \( u = 0 \) at \( x = 0 \), we can use the equation of motion:
\[ v^2 = u^2 + 2as \]
Here, \( a \) is the acceleration, and \( s \) is the distance traveled. Since \( u = 0 \), this simplifies to:
\[ v^2 = 2as \]

3. **Finding Acceleration**:
The acceleration \( a \) can be expressed in terms of the force \( F \) and mass \( m \) of the body:
\[ a = \frac{F}{m} \]
Substituting this into the equation for \( v^2 \):
\[ v^2 = 2 \left(\frac{F}{m}\right) x \]

4. **Finding Velocity**:
Taking the square root of both sides to find \( v \):
\[ v = \sqrt{\frac{2F}{m} x} \]

5. **Substituting Velocity into Power Equation**:
Now, substituting \( v \) back into the power equation:
\[ P = F \cdot v = F \cdot \sqrt{\frac{2F}{m} x} \]

6. **Simplifying Power Expression**:
This can be simplified to:
\[ P = F \sqrt{\frac{2F}{m}} \cdot \sqrt{x} \]
Thus, we can express power as:
\[ P \propto \sqrt{x} \]

7. **Conclusion**:
Therefore, the power delivered by the force \( F \) depends on the position \( x \) as:
\[ P \propto x^{1/2} \]

### Final Answer:
The power delivered by the force \( F \) will depend on position \( x \) as \( P \propto x^{1/2} \).

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