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

Step 1: Know the idea being tested.
This question uses the Heisenberg uncertainty principle. It says we cannot know a particle's exact position and exact momentum at the same time. The rule is $\Delta x \cdot \Delta p \ge \dfrac{h}{4\pi}$, where $\Delta x$ is the uncertainty in position and $\Delta p$ is the uncertainty in momentum. We use the equality to get the smallest uncertainty.
Step 2: Change the mass to kilograms.
Always work in SI units in such problems. The mass is $50$ g. Since $1000$ g $=1$ kg, we get \[ m = \frac{50}{1000} = 0.05 \ \text{kg}. \]
Step 3: Find the uncertainty in speed.
The speed is known to within $2\%$. So the uncertainty in speed is $2\%$ of $50$ m/s. \[ \Delta v = \frac{2}{100}\times 50 = 1 \ \text{m s}^{-1}. \] This is how far the true speed may differ from the measured value.
Step 4: Turn speed uncertainty into momentum uncertainty.
Momentum is $p = mv$, so its uncertainty is $\Delta p = m\,\Delta v$. \[ \Delta p = 0.05 \times 1 = 0.05 \ \text{kg m s}^{-1}. \]
Step 5: Put values into the uncertainty rule.
Using $\Delta x = \dfrac{h}{4\pi\,\Delta p}$ and $h = 6.626\times10^{-34}$, $\pi = 3.14$: \[ \Delta x = \frac{6.626\times10^{-34}}{4\times 3.14 \times 0.05}. \] The bottom is $4\times3.14\times0.05 = 0.628$.
Step 6: Do the final division and conclude.
\[ \Delta x = \frac{6.626\times10^{-34}}{0.628} \approx 1.06\times10^{-33}\ \text{m}. \] So the smallest uncertainty in the ball's position is \[ \boxed{1.06\times10^{-33}\ \text{m}} \]