A point mass performs straight line motion along positive x-axis At t=0 point mass is at point `A(x_(1),0)` it moves such that its velocity is given by `v=(a)/(x)`, where a is positive constant and x is the x-coordinate of position vector mass at a certain time t. Find the time required to move from A to B `(x_(2),0)`

Options

A. `(x_(2)^(2)-x_(1)^(2))/(2a)`
B. `(x_(2)^(2)-x_(1)^(2))/(a)`
C. `(2x_(2)^(2)-x_(1)^(2))/(2a)`
D. `(2x_(2)^(2)-x_(1)^(2))/(a)`

Explanation

To solve the problem, we need to find the time required for a point mass to move from point A at \( x_1 \) to point B at \( x_2 \) along the positive x-axis, given that its velocity is defined by the equation \( v = \frac{a}{x} \), where \( a \) is a positive constant.

### Step-by-Step Solution:

1. **Understand the Relationship Between Velocity and Position**:
The velocity \( v \) can be expressed in terms of position \( x \) and time \( t \) as:
\[ v = \frac{dx}{dt} \]
Given that \( v = \frac{a}{x} \), we can equate the two expressions:
\[ \frac{dx}{dt} = \frac{a}{x} \]

2. **Rearranging the Equation**:
We can rearrange the equation to separate variables:
\[ x \, dx = a \, dt \]

3. **Integrating Both Sides**:
Now, we integrate both sides. The left side will be integrated with respect to \( x \) from \( x_1 \) to \( x_2 \), and the right side will be integrated with respect to \( t \) from \( 0 \) to \( t \):
\[ \int_{x_1}^{x_2} x \, dx = \int_{0}^{t} a \, dt \]

4. **Calculating the Integrals**:
The left side integral is:
\[ \int x \, dx = \frac{x^2}{2} \Big|_{x_1}^{x_2} = \frac{x_2^2}{2} - \frac{x_1^2}{2} \]
The right side integral is:
\[ \int a \, dt = at \Big|_{0}^{t} = at \]

5. **Setting the Integrals Equal**:
Now we set the results of the integrals equal to each other:
\[ \frac{x_2^2}{2} - \frac{x_1^2}{2} = at \]

6. **Solving for Time \( t \)**:
Rearranging the equation to solve for \( t \):
\[ at = \frac{x_2^2 - x_1^2}{2} \]
Thus, we can express \( t \) as:
\[ t = \frac{x_2^2 - x_1^2}{2a} \]

### Final Answer:
The time required to move from point A to point B is:
\[ t = \frac{x_2^2 - x_1^2}{2a} \]