Question

A ball has a mass of \(50\,\text{g}\) and a speed of \(50\,\text{m s}^{-1}\). If the speed is measured within an accuracy of \(2\%\), then the uncertainty in its position (in m) is
\[ (h = 6.626 \times 10^{-34}\,\text{J s}, \ \pi = 3.14) \]

• A. \(2.12 \times 10^{-34}\)
• B. \(1.06 \times 10^{-33}\)
• C. \(2.12 \times 10^{-33}\)
• D. \(1.06 \times 10^{-34}\)

Hint:

For numerical problems based on Heisenberg uncertainty principle: \[ \Delta x \Delta p \geq \frac{h}{4\pi} \] always remember: \[ \Delta p = m\Delta v \] and all quantities must be converted into SI units before substitution.

Answer 1

The Correct Option is B

Concept: This question is based on the famous

Heisenberg Uncertainty Principle, which states that it is impossible to simultaneously determine the exact position and exact momentum of a moving particle. The mathematical expression for uncertainty principle is: \[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \] where, \[ \Delta x = \text{uncertainty in position} \] \[ \Delta p = \text{uncertainty in momentum} \] \[ h = \text{Planck's constant} \] Since momentum is given by: \[ p = mv \] therefore, \[ \Delta p = m \Delta v \] We first calculate the uncertainty in velocity and then substitute into the uncertainty relation to determine uncertainty in position.

Step 1: Convert the given mass into SI unit. Given: \[ m = 50\,\text{g} \] Since, \[ 1000\,\text{g} = 1\,\text{kg} \] Thus, \[ m = \frac{50}{1000} = 0.05\,\text{kg} \]

Step 2: Calculate uncertainty in velocity. Speed of the ball: \[ v = 50\,\text{m s}^{-1} \] Accuracy in speed measurement: \[ 2\% \] Therefore, \[ \Delta v = \frac{2}{100} \times 50 \] \[ \Delta v = 1\,\text{m s}^{-1} \] Hence, uncertainty in velocity is: \[ \Delta v = 1\,\text{m s}^{-1} \]

Step 3: Calculate uncertainty in momentum. Using: \[ \Delta p = m\Delta v \] Substituting the values: \[ \Delta p = 0.05 \times 1 \] \[ \Delta p = 0.05\,\text{kg m s}^{-1} \]

Step 4: Apply Heisenberg uncertainty principle. We know: \[ \Delta x \cdot \Delta p = \frac{h}{4\pi} \] Therefore, \[ \Delta x = \frac{h}{4\pi \Delta p} \] Substituting the values: \[ \Delta x = \frac{6.626 \times 10^{-34}} {4 \times 3.14 \times 0.05} \] First calculate denominator: \[ 4 \times 3.14 \times 0.05 = 0.628 \] Thus, \[ \Delta x = \frac{6.626 \times 10^{-34}}{0.628} \] \[ \Delta x = 1.055 \times 10^{-33}\,\text{m} \] Approximating: \[ \Delta x \approx 1.06 \times 10^{-33}\,\text{m} \]

Step 5: Write the final answer. Hence, the uncertainty in position is: \[ \boxed{1.06 \times 10^{-33}\,\text{m}} \] Therefore, the correct option is: \[ \boxed{(B)} \]