Question:medium

In an atom, electron is moving with a speed of $x~ms^{-1}.$ If its speed is measured within an accuracy of 0.001%, what is its uncertainty in position (in m)? $(m_{e}=9\times10^{-31}kg, h=6.6\times10^{-34}Js)$

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Always ensure all units are in the MKS system (meters, kilograms, seconds) before applying the Heisenberg formula to avoid errors in the power of 10. The accuracy percentage must always be converted to a decimal factor (0.001% = 0.00001) before multiplying by the speed.
Updated On: Jun 7, 2026
  • $\frac{3\pi x}{55}$
  • $\frac{55\pi}{3x}$
  • $\frac{55}{3\pi x}$
  • $\frac{55x}{3\pi}$
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The Correct Option is C

Solution and Explanation

Step 1: Pick the right rule.
When we are not sure about both speed and position of a tiny particle, we use the Heisenberg uncertainty principle: $\Delta x\cdot \Delta p\geq \frac{h}{4\pi}$.
Step 2: Bring in the mass.
Momentum is mass times speed, so $\Delta p=m\,\Delta v$. Putting this in gives $\Delta x\geq \frac{h}{4\pi m\,\Delta v}$.
Step 3: Find the uncertainty in speed.
The speed is $x$ and it is known within $0.001\%$. So \[ \Delta v=x\times\frac{0.001}{100}=x\times10^{-5}\ \text{m/s} \]
Step 4: Put every value in.
With $m=9\times10^{-31}$ kg and $h=6.6\times10^{-34}$ Js: \[ \Delta x=\frac{6.6\times10^{-34}}{4\pi(9\times10^{-31})(x\times10^{-5})} \]
Step 5: Tidy the powers of ten.
The denominator constants give $4\times9=36$ and $10^{-31}\times10^{-5}=10^{-36}$. So \[ \Delta x=\frac{6.6\times10^{-34}}{36\pi\,x\times10^{-36}}=\frac{6.6\times10^{2}}{36\pi x}=\frac{660}{36\pi x} \]
Step 6: Reduce the fraction.
Divide top and bottom by 12: $\frac{660}{36}=\frac{55}{3}$. \[ \boxed{\Delta x=\dfrac{55}{3\pi x}\ \text{m}} \]
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