Force acting on a particle of mass m moving in straight line varies with the velocity of the particle as `F=K//V` K is constant then speed of the particle in time t

Options

A. `sqrt((kt)/(m))`
B. `sqrt((2kt)/(m))`
C. `2sqrt((kt)/(m))`
D. none of these

Explanation

To solve the problem, we need to find the speed of a particle of mass \( m \) as a function of time \( t \), given that the force acting on it varies with its velocity \( v \) as \( F = \frac{K}{v} \), where \( K \) is a constant.

### Step-by-Step Solution:

1. **Start with Newton's Second Law**:
According to Newton's second law, the force acting on a particle is equal to the mass of the particle multiplied by its acceleration:
\[ F = m \frac{dv}{dt} \]

2. **Substitute the Given Force**:
We know that the force \( F \) is given by \( F = \frac{K}{v} \). Therefore, we can equate the two expressions for force:
\[ m \frac{dv}{dt} = \frac{K}{v} \]

3. **Rearranging the Equation**:
To separate the variables, we can multiply both sides by \( v \) and rearrange:
\[ v \, dv = \frac{K}{m} \, dt \]

4. **Integrate Both Sides**:
Now we integrate both sides. The left side will be integrated with respect to \( v \) from \( 0 \) to \( v \), and the right side will be integrated with respect to \( t \) from \( 0 \) to \( t \):
\[ \int_0^v v \, dv = \int_0^t \frac{K}{m} \, dt \]

5. **Calculate the Integrals**:
The integral on the left side is:
\[ \frac{1}{2} v^2 \bigg|_0^v = \frac{1}{2} v^2 \]
The integral on the right side is:
\[ \frac{K}{m} t \bigg|_0^t = \frac{K}{m} t \]

6. **Set the Results Equal**:
Setting the results of the integrals equal gives us:
\[ \frac{1}{2} v^2 = \frac{K}{m} t \]

7. **Solve for Velocity \( v \)**:
To find \( v \), we multiply both sides by 2:
\[ v^2 = \frac{2K}{m} t \]
Taking the square root of both sides gives:
\[ v = \sqrt{\frac{2K}{m} t} \]

### Final Result:
The speed of the particle at time \( t \) is:
\[ v(t) = \sqrt{\frac{2K}{m} t} \]