A particle performing simple harmonic motion,
(i)has parabolic velocity-displacement graph.
(ii)has sinusoidal velocity-time graph.
(iii) has ellipticalvelocity-acceleration graph.Choose the correct statements.

Options

A. (ii) and (iii)
B. (i), (ii) and (iii)
C. (i) and (ii)
D. (i) and(iii)

Explanation

To solve the question regarding the characteristics of a particle performing simple harmonic motion (SHM), we need to analyze each statement provided:

### Step-by-Step Solution:

1. **Understanding the Motion**:
- A particle in SHM can be described by the displacement equation:
\[ x(t) = A \sin(\omega t) \]
- Here, \(A\) is the amplitude and \(\omega\) is the angular frequency.

2. **Velocity-Displacement Graph**:
- The velocity \(v\) is given by the derivative of displacement:
\[ v(t) = \frac{dx}{dt} = A \omega \cos(\omega t) \]
- To analyze the velocity-displacement relationship, we can express \(\sin^2(\omega t) + \cos^2(\omega t) = 1\):
\[ \frac{x^2}{A^2} + \frac{v^2}{(A \omega)^2} = 1 \]
- This equation represents an ellipse, not a parabola. Therefore, the statement (i) "has parabolic velocity-displacement graph" is **incorrect**.

3. **Velocity-Time Graph**:
- The velocity as a function of time is:
\[ v(t) = A \omega \cos(\omega t) \]
- The graph of \(v\) versus \(t\) is a cosine function, which is sinusoidal. Thus, statement (ii) "has sinusoidal velocity-time graph" is **correct**.

4. **Velocity-Acceleration Graph**:
- The acceleration \(a\) is given by:
\[ a(t) = \frac{dv}{dt} = -A \omega^2 \sin(\omega t) \]
- We can express this relationship similarly:
\[ \frac{v^2}{(A \omega)^2} + \frac{a^2}{(A \omega^2)^2} = 1 \]
- This also represents an ellipse. Therefore, statement (iii) "has elliptical velocity-acceleration graph" is **correct**.

5. **Conclusion**:
- From the analysis:
- Statement (i) is incorrect.
- Statement (ii) is correct.
- Statement (iii) is correct.
- Thus, the correct statements are (ii) and (iii).

### Final Answer:
The correct statements are (ii) and (iii).

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