Question

A golf ball of mass ‘m’ has a speed of 50 m/s. If the speed can be measured within accuracy of 2\%, the uncertainty in the position is

• A. \( \frac{h}{4\pi \ m} \)
• B. \( \frac{h}{16\pi \ m} \)
• C. \( \frac{h}{4 \pi \ m} \times 10^3 \)
• D. \( \frac{h}{16 \pi \ m} \times 10^3 \)

Hint:

Use the Heisenberg uncertainty principle for problems involving uncertainty in position and momentum.

Answer 1

The Correct Option is C

According to **Heisenberg's Uncertainty Principle**: \[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \] where \( \Delta x \) is the uncertainty in position and \( \Delta p \) is the uncertainty in momentum. The uncertainty in momentum is given by: \[ \Delta p = m \cdot \Delta v \] Given that the velocity is **\( v = 50 \) m/s** with an **uncertainty of 2\%**, we calculate: \[ \Delta v = \frac{2}{100} \times 50 = 1 \text{ m/s} \] Thus, \[ \Delta p = m \times 1 \] From Heisenberg's principle: \[ \Delta x \geq \frac{h}{4\pi \Delta p} \] Substituting \( \Delta p = m \): \[ \Delta x \geq \frac{h}{4\pi m} \] Since the uncertainty in speed is in **meters per second (m/s),** we express the uncertainty in position in **millimeters (mm)**: \[ \Delta x \geq \frac{h}{4\pi m} \times 10^3 \ \text{mm} \] Hence, the correct answer is: \[ \boxed{\frac{h}{4\pi m} \times 10^3} \]