JEE Physics 2025 Question with Solution

Given: Initial position $x_0$ and initial momentum $p_0$ for a simple harmonic motion with given angular frequency $\omega$.

Find: Whether Assertion (A) and Reason (R) are true, and whether (R) correctly explains (A).

For SHM, the displacement and momentum may be written as

$$x(t) = A \cos(\omega t + \phi)$$

and

$$p(t) = m\frac{dx}{dt} = -m\omega A \sin(\omega t + \phi)$$

At $t = 0$,

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

Squaring and adding,

$$x_0^2 + \left(\frac{p_0}{m\omega}\right)^2 = A^2$$

So,

$$A = \sqrt{x_0^2 + \left(\frac{p_0}{m\omega}\right)^2}$$

Also,

$$\tan \phi = -\frac{p_0}{m\omega x_0}$$

Therefore, both the amplitude $A$ and phase $\phi$ can be determined from $x_0$ and $p_0$. Once $A$ and $\phi$ are known, $x(t)$ and $p(t)$ are completely determined for any time $t$.

Hence, Assertion (A) is true, Reason (R) is true, and Reason (R) correctly explains Assertion (A).

Therefore, the correct option is A.

Given: Assertion (A): Knowing the 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 given angular frequency $\omega$.

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

Find: The correct option.

Step 1: Write the standard SHM form:

$$x(t) = A \cos(\omega t + \phi)$$ $$p(t) = -m\omega A \sin(\omega t + \phi)$$

Step 2: Apply the initial condition $t = 0$:

$$x_0 = A \cos \phi$$ $$p_0 = -m\omega A \sin \phi$$

Step 3: Determine the amplitude by squaring and adding:

$$x_0^2 + \left(\frac{p_0}{m\omega}\right)^2 = A^2$$

Thus,

$$A = \sqrt{x_0^2 + \left(\frac{p_0}{m\omega}\right)^2}$$

Step 4: Determine the phase:

$$\tan \phi = -\frac{p_0}{m\omega x_0}$$

So $\phi$ is also determined by $x_0$ and $p_0$.

Step 5: Since the complete SHM is specified by amplitude $A$ and phase $\phi$, knowing $x_0$ and $p_0$ is sufficient to determine both $x(t)$ and $p(t)$ at any time $t$.

Therefore, both (A) and (R) are true, and (R) is the correct explanation of (A).

The correct option is A.