The uncertainties in the velocities of two particles A and B are 0.05 and $0.02\text{m}.{\sec }^{-1}$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

Heisenberg uncertainty principle,

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

From given data ,

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

$v_B=5m.{\sec }^{-1}$

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

$\frac{\Delta x_{\text{A}}}{\Delta x_{\text{B}}}=\frac{{\text{m}}_{\text{B}}}{{\text{m}}_{\text{A}}}\times \frac{\Delta {\text{v}}_{\text{B}}}{\Delta {\text{v}}_{\text{A}}}$                  

$\therefore \frac{\Delta x_{\text{A}}}{\Delta x_{\text{B}}}=\frac{\text{5m}}{\text{m}}\times \frac{0.02}{0.05}=2$