The maximum acceleration of a simple harmonic oscillator is `a_(0)` and maximum velocity is `v_(0)`. What is the amplitude?

Options

A. `(v_(0)^(2))/(a_(0))`
B. `a_(0)v_(0)`
C. `(a_(0)^(2))/(v_(0))`
D. `(1)/(a_(0)v_(0))`

Explanation

To find the amplitude of a simple harmonic oscillator given its maximum acceleration \( a_0 \) and maximum velocity \( v_0 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationships**:
- The maximum velocity \( v_{\text{max}} \) in simple harmonic motion (SHM) is given by:
\[ v_{\text{max}} = A \omega \]
where \( A \) is the amplitude and \( \omega \) is the angular frequency.

2. **Maximum Acceleration**:
- The maximum acceleration \( a_{\text{max}} \) in SHM is given by:
\[ a_{\text{max}} = A \omega^2 \]

3. **Express \( \omega \)**:
- From the maximum velocity equation, we can express \( \omega \) in terms of \( v_{\text{max}} \) and \( A \):
\[ \omega = \frac{v_{\text{max}}}{A} \]

4. **Substitute \( \omega \) into the Acceleration Equation**:
- Substitute \( \omega \) into the maximum acceleration equation:
\[ a_{\text{max}} = A \left(\frac{v_{\text{max}}}{A}\right)^2 \]
- This simplifies to:
\[ a_{\text{max}} = A \frac{v_{\text{max}}^2}{A^2} \]
- Further simplifying gives:
\[ a_{\text{max}} = \frac{v_{\text{max}}^2}{A} \]

5. **Rearranging for Amplitude \( A \)**:
- Rearranging the equation to solve for \( A \):
\[ A = \frac{v_{\text{max}}^2}{a_{\text{max}}} \]

6. **Substituting Given Values**:
- Given that \( v_{\text{max}} = v_0 \) and \( a_{\text{max}} = a_0 \), we can write:
\[ A = \frac{v_0^2}{a_0} \]

### Final Answer:
Thus, the amplitude \( A \) is given by:
\[ A = \frac{v_0^2}{a_0} \]