The uncertainties in the velocities of two particles A and B are 0.05 and 0.02m.sec−1respectively. The mass of B is five times to that of mass A. What is the ratio of uncertainties (ΔxAΔxB) in their positions

# The uncertainties in the velocities of two particles A and B are 0.05 and $0.02\text{m}.{\mathrm{sec}}^{-1}$respectively. The mass of B is five times to that of mass A. What is the ratio of uncertainties $\left(\frac{\Delta {x}_{\text{A}}}{\Delta {x}_{\text{B}}}\right)$ in their positions

1. A

2

2. B

0.25

3. C

4

4. D

1

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### Solution:

Heisenberg uncertainty principle,

$\Delta x.\Delta \text{v}=\frac{\text{h}}{4\pi \text{m}}$

From given data ,

${\text{v}}_{\text{A}}=0.05\text{\hspace{0.17em}m}.{\mathrm{sec}}^{-1}$

${v}_{B}=5m.{\mathrm{sec}}^{-1}$

Mass of A is equal to ${m}_{\text{A}}=m$

Then mass of B is equal to 5 times than mass of A.

So, ${m}_{\text{B}}=5m$

The ratio of uncertainties of A and B is

$\frac{\Delta {x}_{\text{A}}}{\Delta {x}_{\text{B}}}=\frac{{\text{m}}_{\text{B}}}{{\text{m}}_{\text{A}}}×\frac{\Delta {\text{v}}_{\text{B}}}{\Delta {\text{v}}_{\text{A}}}$

$\therefore \frac{\Delta {x}_{\text{A}}}{\Delta {x}_{\text{B}}}=\frac{\text{5m}}{\text{m}}×\frac{0.02}{0.05}=2$

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