The power supplied by a force acting on a particle moving in a straight line is constant. The velocity of the particle varies with displacement as `x^(1/K)`. Find the value of K.

Answer

To solve the problem, we need to analyze the relationship between power, force, and the velocity of a particle moving in a straight line, given that the velocity varies with displacement as \( x^{(1/K)} \).

### Step-by-Step Solution:

1. **Understanding Power**:
Power (P) is defined as the work done per unit time. Mathematically, it can be expressed as:
\[ P = \frac{W}{t} = F \cdot v \]
where \( F \) is the force acting on the particle and \( v \) is its velocity.

2. **Expressing Force**:
The force can also be expressed using Newton's second law:
\[ F = m \cdot a \]
where \( m \) is the mass of the particle and \( a \) is its acceleration.

3. **Relating Acceleration and Velocity**:
We know that acceleration \( a \) can be expressed in terms of velocity \( v \) and displacement \( s \) using the kinematic equation:
\[ a = \frac{dv}{dt} = \frac{dv}{ds} \cdot \frac{ds}{dt} = \frac{dv}{ds} \cdot v \]
Therefore, we can write:
\[ a = v \frac{dv}{ds} \]

4. **Substituting into Power Equation**:
Substituting \( a \) into the power equation gives:
\[ P = m \cdot (v \frac{dv}{ds}) \cdot v = m v^2 \frac{dv}{ds} \]

5. **Constant Power**:
Since the power \( P \) is constant, we can express it as:
\[ P = m v^2 \frac{dv}{ds} = \text{constant} \]

6. **Rearranging the Equation**:
Rearranging the equation gives:
\[ v^2 \frac{dv}{ds} = \frac{P}{m} = \text{constant} \]

7. **Integrating**:
Let \( C = \frac{P}{m} \) (a constant). We can write:
\[ v^2 dv = C ds \]
Integrating both sides:
\[ \int v^2 dv = \int C ds \]
This results in:
\[ \frac{v^3}{3} = Cs + C_1 \]
where \( C_1 \) is the constant of integration.

8. **Expressing Velocity in Terms of Displacement**:
Rearranging gives:
\[ v^3 = 3Cs + 3C_1 \]
This shows that \( v^3 \) is proportional to \( s \):
\[ v^3 \propto s \]

9. **Relating to Given Expression**:
We are given that \( v \propto s^{(1/K)} \). From our result, we have:
\[ v \propto s^{1/3} \]
This implies:
\[ \frac{1}{K} = \frac{1}{3} \implies K = 3 \]

### Final Answer:
The value of \( K \) is \( 3 \).