Question

The uncertainty in the determination of the position of a small ball of mass 10 g is \(10^{-33}\) m. With what \% of accuracy can its speed be measured, if it has a speed of 52.5 m s\(^{-1}\)?

• A. \( 1.0\% \)
• B. \( 20\% \)
• C. \( 10\% \)
• D. \( 2.0\% \)
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Hint:

Heisenberg's uncertainty principle states that the product of uncertainties in position and momentum must be greater than or equal to \( \frac{h}{4\pi} \).

Answer 1

The Correct Option is C

Step 1: Applying Heisenberg's Uncertainty Principle \[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \] where \( h = 6.6 \times 10^{-34} \) Js, \( \Delta x = 10^{-33} \) m, and mass \( m = 10 \) g \( = 0.01 \) kg. Step 2: Calculating Uncertainty in Momentum \[ \Delta p \geq \frac{6.6 \times 10^{-34}}{4\pi \times 10^{-33}} \] \[ \Delta p \geq \frac{6.6}{12.56} \times 10^{-1} = 0.525 \times 10^{-1} = 0.0525 \text{ kg m/s} \] Step 3: Finding Velocity Uncertainty \[ \Delta v = \frac{\Delta p}{m} = \frac{0.0525}{0.01} = 5.25 \text{ m/s} \] Step 4: Finding Percentage Accuracy \[ \text{Percentage Accuracy} = \left( \frac{\Delta v}{v} \right) \times 100 \] \[ = \left( \frac{5.25}{52.5} \right) \times 100 = 10\% \] \bigskip