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The velocity of a proton in an accelerator is known to an accuracy of 0.250% of the speed of light. (This could be small compared with its velocity.) What is the smallest possible uncertainty in its position?
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Final Answer
$42.1 \textrm{ fm}$
Solution Video

OpenStax College Physics Solution, Chapter 29, Problem 65 (Problems & Exercises) (1:18)

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Video Transcript

This is College Physics Answers with Shaun Dychko. We know the uncertainty in the velocity of a proton that is 0.0025 times the speed of light and the mass of a proton, we know, is 1.6726 times 10 to the minus 27 kilograms and we'll assume that this is known with no uncertainty. The Heisenberg uncertainty principle; equation [29.43], says that the uncertainty in position mutliplied by uncertainty in momentum has to be more than or equal to Planck's constant divided by 4<i>π</i>. And we'll assume that the uncertainty in momentum can be attributed entirely to the uncertainty in velocity assuming that we know mass perfectly, in which case, <i>ΔP</i> will equal <i>m Δv</i> and we'll substitute that in place of <i>Δp</i> here. And let's solve this for <i>Δx</i> by dividing both sides by <i>m Δv</i>. And we get that the uncertainty in position then is more than or equal to Planck's constant over 4<i>π</i> times mass times uncertainty in velocity. So that's Planck's constant divided by 4<i>π</i> times the mass of a proton times 0.0025 times the speed of light, which at a minimum then the uncertainty in position is 42.1 femtometers.