The rate of doing work by force acting on a particle moving along x-axis depends on position x of particle and is equal to 2x. The velocity of particle is given by expression :-

Options

A. `[(3x^(2))/(m)]^(1//3)`
B. `[(3x^(2))/(2m)]^(1//3)`
C. `((2mx)/(9))^(1//2)`
D. `[(mx^(2))/(3)]^(1//2)`

Answer

To find the velocity of a particle moving along the x-axis where the rate of doing work by the force is given as \( P = 2x \), we can follow these steps:

### Step 1: Understand the relationship between power, force, and velocity
The rate of doing work (power) is given by the formula:
\[ P = F \cdot v \]
where \( F \) is the force acting on the particle and \( v \) is its velocity.

### Step 2: Set up the equation using the given power
From the problem, we know that:
\[ P = 2x \]
Thus, we can write:
\[ F \cdot v = 2x \]

### Step 3: Relate force to mass and acceleration
According to Newton's second law, the force can also be expressed as:
\[ F = m \cdot a \]
where \( m \) is the mass of the particle and \( a \) is its acceleration. We can express acceleration in terms of velocity and position:
\[ a = \frac{dv}{dt} = v \frac{dv}{dx} \]
Substituting this into the force equation gives:
\[ F = m \cdot v \frac{dv}{dx} \]

### Step 4: Substitute force into the power equation
Now we can substitute this expression for force back into the power equation:
\[ m \cdot v \frac{dv}{dx} \cdot v = 2x \]
This simplifies to:
\[ m v^2 \frac{dv}{dx} = 2x \]

### Step 5: Rearrange the equation for integration
Rearranging gives:
\[ v^2 dv = \frac{2}{m} x dx \]

### Step 6: Integrate both sides
Now we integrate both sides:
\[ \int v^2 dv = \frac{2}{m} \int x dx \]
This results in:
\[ \frac{v^3}{3} = \frac{2}{m} \cdot \frac{x^2}{2} + C \]
The constant \( C \) can be determined based on initial conditions, but for this problem, we can assume \( C = 0 \) for simplicity.

### Step 7: Simplify the equation
This simplifies to:
\[ \frac{v^3}{3} = \frac{x^2}{m} \]
Multiplying both sides by 3 gives:
\[ v^3 = \frac{3x^2}{m} \]

### Step 8: Solve for velocity
Taking the cube root of both sides, we find:
\[ v = \left(\frac{3x^2}{m}\right)^{\frac{1}{3}} \]

### Final Answer
Thus, the velocity of the particle is given by:
\[ v = \left(\frac{3}{m}\right)^{\frac{1}{3}} x^{\frac{2}{3}} \]

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