Two bodies `P` and `Q` of equal masses are suspended from two separate massless springs of force constants `k_(1)` and `k_(2)` respectively. If the two bodies oscillate vertically such that their maximum velocities are equal. The ratio of the amplitude of `P` to that of `Q` is

Options

A. `sqrt((k_(2))/(k_(1))`
B. `(k_(1))/(k_(2))`
C. `sqrt(k_(2))/(k_(1))`
D. `(k_(2))/(k_(1))`

Explanation

To solve the problem, we need to find the ratio of the amplitudes of two bodies P and Q suspended from springs with different force constants, given that their maximum velocities are equal.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two bodies P and Q with equal masses (let's denote the mass as `m`).
- They are attached to springs with spring constants `k1` (for P) and `k2` (for Q).
- The maximum velocities of both bodies during oscillation are equal.

2. **Setting Up the Equations**:
- The maximum velocity in simple harmonic motion (SHM) can be expressed as:
\[ v_{max} = \omega A \]
where \( \omega \) is the angular frequency and \( A \) is the amplitude of oscillation.

3. **Expressing Angular Frequency**:
- The angular frequency \( \omega \) for a mass-spring system is given by:
\[ \omega = \sqrt{\frac{k}{m}} \]
- Therefore, for body P:
\[ \omega_1 = \sqrt{\frac{k_1}{m}} \]
- And for body Q:
\[ \omega_2 = \sqrt{\frac{k_2}{m}} \]

4. **Equating Maximum Velocities**:
- Since the maximum velocities of P and Q are equal:
\[ v_P = v_Q \]
- Substituting the expressions for maximum velocities:
\[ \omega_1 A_1 = \omega_2 A_2 \]
- Replacing \( \omega_1 \) and \( \omega_2 \):
\[ \sqrt{\frac{k_1}{m}} A_1 = \sqrt{\frac{k_2}{m}} A_2 \]

5. **Simplifying the Equation**:
- Since the masses are equal, we can cancel \( m \) from both sides:
\[ \sqrt{k_1} A_1 = \sqrt{k_2} A_2 \]

6. **Finding the Ratio of Amplitudes**:
- Rearranging the equation gives:
\[ \frac{A_1}{A_2} = \frac{\sqrt{k_2}}{\sqrt{k_1}} \]
- Thus, the ratio of the amplitudes of P to Q is:
\[ \frac{A_1}{A_2} = \sqrt{\frac{k_2}{k_1}} \]

### Final Answer:
The ratio of the amplitude of P to that of Q is:
\[ \frac{A_1}{A_2} = \sqrt{\frac{k_2}{k_1}} \]