A particle of mass `m` is driven by a machine that delivers a constant power `k` watts. If the particle starts from rest the force on the particle at time `t` is

Options

A. `sqrt((mk)/(2))t^(-1//2)`
B. `sqrt(mk)" t"^(-1//2)`
C. `sqrt(2mk)" t"^(-1//2)`
D. `(1)/(2)sqrt(mk)" t"^(-1//2)`

Explanation

To solve the problem, we need to find the force acting on a particle of mass \( m \) that is driven by a machine delivering a constant power \( k \) watts. The particle starts from rest.

### Step-by-step Solution:

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

2. **Setting Up the Equation**:
Since the power is constant and equal to \( k \):
\[ F \cdot v = k \]
Rearranging gives:
\[ F = \frac{k}{v} \]

3. **Relating Velocity and Acceleration**:
The acceleration \( a \) of the particle can be expressed as:
\[ a = \frac{dv}{dt} \]
Therefore, we can write the force in terms of mass and acceleration:
\[ F = m \cdot a = m \cdot \frac{dv}{dt} \]

4. **Substituting for Force**:
From the previous equations, we have:
\[ m \cdot \frac{dv}{dt} = \frac{k}{v} \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ m \cdot v \cdot dv = k \cdot dt \]

6. **Integrating**:
Now we integrate both sides. The left side integrates with respect to \( v \) and the right side with respect to \( t \):
\[ \int m \cdot v \, dv = \int k \, dt \]
This gives:
\[ \frac{m}{2} v^2 = kt + C \]
where \( C \) is the constant of integration.

7. **Finding the Constant**:
Since the particle starts from rest, when \( t = 0 \), \( v = 0 \):
\[ \frac{m}{2} (0)^2 = k(0) + C \implies C = 0 \]
Thus, we have:
\[ \frac{m}{2} v^2 = kt \]

8. **Solving for Velocity**:
Rearranging gives:
\[ v^2 = \frac{2kt}{m} \]
Therefore:
\[ v = \sqrt{\frac{2kt}{m}} \]

9. **Finding the Force**:
Now substituting \( v \) back into the force equation:
\[ F = \frac{k}{v} = \frac{k}{\sqrt{\frac{2kt}{m}}} \]
This simplifies to:
\[ F = \frac{k \sqrt{m}}{\sqrt{2kt}} = \frac{\sqrt{km}}{\sqrt{2t}} \]

10. **Final Expression for Force**:
Thus, the force on the particle at time \( t \) is:
\[ F = \frac{\sqrt{km}}{\sqrt{2t}} \]