1. Write the general SHM equation

For a simple harmonic oscillator of angular frequency $\omega$, $x(t)=A\cos(\omega t+\phi).$

Its velocity is $v(t)=\frac{dx}{dt}=-A\omega \sin(\omega t+\phi),$ so momentum is $p(t)=mv(t)=-m\omega A\sin(\omega t+\phi).$

2. Use initial conditions

At $t=0$, $x_0=A\cos\phi,$ $p_0=-m\omega A\sin\phi.$

These are two equations in the two unknowns $A$ and $\phi$.

3. Express amplitude in terms of $x_0$ and $p_0$

Squaring and adding, $x_0^2+\left(\frac{p_0}{m\omega}\right)^2=A^2(\cos^2\phi+\sin^2\phi)=A^2.$

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

4. Express phase in terms of $x_0$ and $p_0$

From $x_0=A\cos\phi,\qquad p_0=-m\omega A\sin\phi,$ we get $\cos\phi=\frac{x_0}{A},\qquad \sin\phi=-\frac{p_0}{m\omega A}.$

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

5. Determine motion at any time

Since both $A$ and $\phi$ are known from $x_0$ and $p_0$, we can write $x(t)=A\cos(\omega t+\phi),$ $p(t)=-m\omega A\sin(\omega t+\phi).$

Therefore, knowing $x_0$ and $p_0$ is sufficient to determine position and momentum at any later time $t$.

6. Evaluate Assertion and Reason

• Assertion (A): True, because initial position and momentum fix the full SHM uniquely for given $\omega$.
• Reason (R): True, because amplitude and phase are indeed expressible in terms of $x_0$ and $p_0$.
• Also, (R) correctly explains (A), since once amplitude and phase are known, the entire motion is known.

Therefore, the correct option is $\boxed{D}.$