Two springs, of spring constants `k_(1)` and `K_(2)`, have equal highest velocities, when executing SHM. Then, the ratio of their amplitudes (given their masses are in the ratio `1:2`) will be

Options

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

Explanation

To solve the problem step by step, we need to analyze the relationship between the spring constants, masses, and amplitudes in the context of simple harmonic motion (SHM).

### Step-by-Step Solution:

1. **Understanding Maximum Velocity in SHM**:
In simple harmonic motion, the maximum kinetic energy (KE) is given by:
\[ KE_{max} = \frac{1}{2} m v_{max}^2 \]
The total mechanical energy (E) in SHM is given by:
\[ E = \frac{1}{2} k a^2 \]
where \( k \) is the spring constant and \( a \) is the amplitude.

2. **Setting Up the Equations**:
For the first spring with spring constant \( k_1 \) and mass \( m_1 \):
\[ \frac{1}{2} m_1 v_{max}^2 = \frac{1}{2} k_1 a_1^2 \]
For the second spring with spring constant \( k_2 \) and mass \( m_2 \):
\[ \frac{1}{2} m_2 v_{max}^2 = \frac{1}{2} k_2 a_2^2 \]

3. **Equating the Maximum Velocities**:
Since both springs have equal maximum velocities, we can set the two equations equal to each other:
\[ m_1 v_{max}^2 = k_1 a_1^2 \]
\[ m_2 v_{max}^2 = k_2 a_2^2 \]

4. **Dividing the Two Equations**:
Dividing the first equation by the second gives:
\[ \frac{m_1}{m_2} = \frac{k_1 a_1^2}{k_2 a_2^2} \]

5. **Substituting the Mass Ratio**:
We are given that the masses are in the ratio \( m_1 : m_2 = 1 : 2 \), which means:
\[ \frac{m_1}{m_2} = \frac{1}{2} \]
Substituting this into the equation:
\[ \frac{1}{2} = \frac{k_1 a_1^2}{k_2 a_2^2} \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ a_1^2 = \frac{k_2}{2 k_1} a_2^2 \]

7. **Finding the Ratio of Amplitudes**:
Taking the square root of both sides, we find:
\[ \frac{a_1}{a_2} = \sqrt{\frac{k_2}{2 k_1}} \]

### Final Result:
Thus, the ratio of the amplitudes \( \frac{a_1}{a_2} \) is:
\[ \frac{a_1}{a_2} = \sqrt{\frac{k_2}{2 k_1}} \]