Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
**Assertion (A):** Knowing initial position $x_0$ 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 $\omega$.
**Reason (R):** The amplitude and phase can be expressed in terms of $x_0$ and $p_0$.
In the light of the above statements, choose the correct answer from the options given below:
Answer: (D) Both (A) and (R) are true and (R) is the correct explanation of (A)
SHM: $x = A\sin(\omega t + \phi)$, $p = m\omega A\cos(\omega t + \phi)$. At $t = 0$:
$$x_0 = A\sin\phi, \quad p_0 = m\omega A\cos\phi \Rightarrow A = \sqrt{x_0^2 + \frac{p_0^2}{m^2\omega^2}}, \quad \tan\phi = \frac{m\omega x_0}{p_0}$$
So (R) is true. Once $A$ and $\phi$ are known (with $\omega$ given), $x(t)$ and $p(t)$ are fixed for all time, which is exactly why (A) is true. (R) explains (A).
Solution by Sreeraj P, M.Sc Physics