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# A proton is accelerated to one tenth of the velocity of light. Suppose velocity can be measured with a precision of $\pm1\%$, what can be the uncertainity in its position?

$(a)\;1.05\times10^{-10}m\qquad(b)\;2.3\times10^{-13}m\qquad(c)\;1.3\times10^{-9}m\qquad(d)\;1.05\times10^{-13}m$

Velocity = $\large\frac{3\times10^8}{10}$
$=3\times10^7$m/sec
$\bigtriangleup V = V\times \large\frac{1}{100} = 3\times 10^5$
$\bigtriangleup x .\bigtriangleup V \approx \large\frac{h}{4\pi m}$
$\bigtriangleup x = \large\frac{h}{4\pi m\bigtriangleup V}$
$=\large\frac{6.626\times10^{-34}}{4\pi\times1.673\times10^{-27}\times3\times10^5}$
$=1.05\times10^{-13}$m