A partical of mass `m` is driven by a machine that deleveres a constant power `k` watts. If the partical starts from rest the force on the partical 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. `1/2sqrt(mk)t^(-1//2)`

Explanation

To find the force on a particle of mass \( m \) that is driven by a machine delivering a constant power \( k \) watts, we can follow these steps:

### Step 1: Understand the relationship between power, work, and kinetic energy.
The power delivered by the machine is defined as the rate of doing work. Mathematically, this can be expressed as:
\[ P = \frac{W}{t} \]
where \( W \) is the work done and \( t \) is the time. Given that the power is constant and equal to \( k \), we can write:
\[ W = k \cdot t \]

### Step 2: Apply the work-energy theorem.
According to the work-energy theorem, the work done on the particle is equal to the change in kinetic energy. Since the particle starts from rest, its initial kinetic energy is zero. Therefore, the work done can be expressed as:
\[ W = \Delta KE = \frac{1}{2} mv^2 - 0 = \frac{1}{2} mv^2 \]

### Step 3: Set the expressions for work equal to each other.
From the previous steps, we have:
\[ k \cdot t = \frac{1}{2} mv^2 \]
Rearranging this gives:
\[ mv^2 = 2kt \]

### Step 4: Solve for velocity \( v \).
Dividing both sides by \( m \), we find:
\[ v^2 = \frac{2kt}{m} \]
Taking the square root of both sides, we get:
\[ v = \sqrt{\frac{2kt}{m}} \]

### Step 5: Find the acceleration \( a \).
Acceleration is the derivative of velocity with respect to time:
\[ a = \frac{dv}{dt} \]
Using the expression for \( v \):
\[ a = \frac{d}{dt} \left( \sqrt{\frac{2kt}{m}} \right) \]
Using the chain rule:
\[ a = \frac{1}{2} \left( \frac{2k}{m} \right)^{1/2} t^{-1/2} \]
This simplifies to:
\[ a = \sqrt{\frac{2k}{m}} \cdot \frac{1}{2} t^{-1/2} \]

### Step 6: Calculate the force \( F \).
Using Newton's second law, the force can be expressed as:
\[ F = m \cdot a \]
Substituting the expression for \( a \):
\[ F = m \cdot \left( \sqrt{\frac{2k}{m}} \cdot \frac{1}{2} t^{-1/2} \right) \]
This simplifies to:
\[ F = \frac{m}{2} \cdot \sqrt{\frac{2k}{m}} \cdot t^{-1/2} \]
Further simplifying gives:
\[ F = \frac{1}{2} \sqrt{2km} \cdot t^{-1/2} \]

### Final Expression for Force
Thus, the force on the particle at time \( t \) is:
\[ F = \frac{1}{2} \sqrt{\frac{2k}{m}} \cdot t^{-1/2} \]