Q60 / 75 · JEE Main 2025 (28 Jan Shift 2) · Physics

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 $\mathrm{p}_0$. In the light of the above statements, choose the correct answer from the options given below :

  • A
    (A) is false but (R) is true
  • B
    (A) is true but (R) is false
  • C
    Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
  • Correct answer
    D
    Both $(\mathbf{A})$ and $(\mathbf{R})$ are true and $(\mathbf{R})$ is the correct explanation of $(\mathbf{A})$

Solution

$\begin{aligned} & \text { If we express position } x(t)=A \sin (\omega t+\phi) \\ & \text { then } x_0=A \sin \phi \\ & v_0=A \omega \cos \phi \\ & \Rightarrow \tan \phi=\frac{\omega x_0}{v_0} \\ & A=\sqrt{x_0^2+\frac{v_0^2}{\omega^2}}\end{aligned}$ Hence both position and linear momentum of a particle can be expressed as a function of time if we know initial momentum and position

Hence, the correct answer is option: 'D'.

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