Given below are two statements: Statement (I): It is impossible to specify simultaneously, with arbitrary precision,…

🧠 The "Aha!" Moment Heisenberg's Uncertainty Principle is the "speed limit" of information in the universe. It isn't a failure of our tools, but a fundamental property of nature. Statement I is the direct law. Statement II is a specific mathematical case where we try to balance the scales perfectly between position and momentum.

🗺️ The Proof for Statement II

1. The Principle: $\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$
2. The Condition: If uncertainties in position and momentum are equal ($\Delta x = \Delta p$): $(\Delta p)^2 \ge \frac{h}{4\pi} \implies \Delta p \ge \sqrt{\frac{h}{4\pi}} = \frac{1}{2} \sqrt{\frac{h}{\pi}}$
3. Link to Velocity: Since momentum uncertainty is $\Delta p = m \Delta v$: $m \Delta v \ge \frac{1}{2} \sqrt{\frac{h}{\pi}} \implies \Delta v \ge \frac{1}{2m} \sqrt{\frac{h}{\pi}}$ Both statements are mathematically and conceptually flawless.

⚡ The Equal Uncertainty Trap If you ever see a problem where $\Delta x = \Delta p$, remember that the uncertainty in velocity will always involve a square root of Planck's constant ($h$). It’s a common "numerical shortcut" in competitive physics.

⚠️ Watch the Pi! $\sqrt{h/4\pi}$ simplifies to $\frac{1}{2}\sqrt{h/\pi}$. Don't let the placement of the $1/2$ throw you off; it's simply the square root of the $1/4$ that sits inside the original principle.

$\boxed{\text{Answer: (b)}}$