A particle executing harmonic motion is having velocities `v_1` and `v_2` at distances is `x_1` and `x_2` from the equilibrium position. The amplitude of the motion is

Options

A. `sqrt((v_1^2+x_2-v_2^2x_1)/(v_1^2-v_2^2))`
B. `sqrt((v_1^2x_1^2-v_2^2x_2^2)/(v_1^2+v_2^2))`
C. `sqrt((v_1^2x_2^2-v_2^2x_1^2)/(v_1^2-v_2^2))`
D. `sqrt((v_1^2x_2^2+v_2^2x_1^2)/(v_1^2+v_2^2))`

Explanation

To solve the problem of finding the amplitude of a particle executing simple harmonic motion (SHM) given its velocities \( v_1 \) and \( v_2 \) at distances \( x_1 \) and \( x_2 \) from the equilibrium position, we can follow these steps:

### Step 1: Understand the velocity formula in SHM
The velocity \( v \) of a particle in SHM is given by the formula:
\[ v = \omega \sqrt{A^2 - x^2} \]
where:
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude,
- \( x \) is the displacement from the equilibrium position.

### Step 2: Write the equations for the given velocities
For the two positions \( x_1 \) and \( x_2 \), we can write:
\[ v_1 = \omega \sqrt{A^2 - x_1^2} \]
\[ v_2 = \omega \sqrt{A^2 - x_2^2} \]

### Step 3: Square both equations
Squaring both equations gives:
\[ v_1^2 = \omega^2 (A^2 - x_1^2) \]
\[ v_2^2 = \omega^2 (A^2 - x_2^2) \]

### Step 4: Rearranging the equations
From the first equation, we can express \( A^2 \):
\[ A^2 = \frac{v_1^2}{\omega^2} + x_1^2 \]
From the second equation:
\[ A^2 = \frac{v_2^2}{\omega^2} + x_2^2 \]

### Step 5: Set the two expressions for \( A^2 \) equal to each other
Equating the two expressions for \( A^2 \):
\[ \frac{v_1^2}{\omega^2} + x_1^2 = \frac{v_2^2}{\omega^2} + x_2^2 \]

### Step 6: Rearranging to isolate \( A^2 \)
Rearranging gives:
\[ \frac{v_1^2 - v_2^2}{\omega^2} = x_2^2 - x_1^2 \]

### Step 7: Solve for \( A^2 \)
Now, we can express \( A^2 \) in terms of \( v_1 \), \( v_2 \), \( x_1 \), and \( x_2 \):
\[ A^2 = \frac{v_1^2 x_2^2 - v_2^2 x_1^2}{v_1^2 - v_2^2} \]

### Step 8: Take the square root to find \( A \)
Finally, taking the square root gives us the amplitude \( A \):
\[ A = \sqrt{\frac{v_1^2 x_2^2 - v_2^2 x_1^2}{v_1^2 - v_2^2}} \]

### Conclusion
Thus, the amplitude of the motion is:
\[ A = \sqrt{\frac{v_1^2 x_2^2 - v_2^2 x_1^2}{v_1^2 - v_2^2}} \]