Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Knowing initial position $$x_{\circ}$$ and initial momentum $$p_{\circ}$$ is enough to determine the position and momentum at any time $$t$$ for a simple harmonic motion with a given angular frequency $$\omega$$. Reason (R): The amplitude and phase can be expressed in terms of $$x_{\circ}$$ and $$p_{\circ}$$. In the light of the above statements, choose the correct answer from the options given below :

Assertion (A): Knowing initial position $$x_0$$ and initial momentum $$p_0$$ is enough to determine the position and momentum at any time $$t$$ for a simple harmonic motion with a given angular frequency $$\omega$$.

This is true. For SHM, $$x(t) = A\sin(\omega t + \phi)$$ and $$p(t) = m\omega A\cos(\omega t + \phi)$$. The two unknowns are the amplitude $$A$$ and the phase $$\phi$$. Given $$x_0$$ and $$p_0$$ at $$t = 0$$, we can determine both $$A$$ and $$\phi$$ uniquely (with $$\omega$$ and $$m$$ known). Therefore, the motion is completely determined for all future times.

Reason (R): The amplitude and phase can be expressed in terms of $$x_0$$ and $$p_0$$.

This is true. From initial conditions:

$$x_0 = A\sin\phi, \quad p_0 = m\omega A\cos\phi$$

$$A^2 = x_0^2 + \frac{p_0^2}{m^2\omega^2}, \quad \tan\phi = \frac{m\omega x_0}{p_0}$$

Moreover, R directly explains A — the reason we can determine the motion from $$x_0$$ and $$p_0$$ is precisely because we can express the amplitude and phase in terms of these initial values.

The answer is Option D: Both (A) and (R) are true and (R) is the correct explanation of (A).