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The mass of a proton is $1.67\times 10^{-27}\textrm{ kg}$. If a proton has the same momentum as a photon with a wavelength of 325 nm, what is its speed?
  1. $2.73\times 10^{-3}\textrm{ m/s}$
  2. $0.819\textrm{ m/s}$
  3. $1.22 \textrm{ m/s}$
  4. $2.71\times 10^{4}\textrm{ m/s}$
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OpenStax College Physics for AP® Courses Solution, Chapter 29, Problem 7 (Test Prep for AP® Courses) (0:44)

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This is College Physics Answers with Shaun Dychko. The momentum of a photon is Planck's constant divided by its wavelength— that's equation [29.22]. The momentum of a proton moving at a non-relativistic speed is its mass times its velocity. So let's equate these two things— <i>mv</i> and <i>h</i> over <i>λ</i> to figure out what the speed of the proton would be in order to have the same momentum as this photon of wavelength 325 nanometers. So we'll divide both sides by <i>m</i> here and we get that the speed then is Planck's constant over <i>m</i> times the wavelength. So that's Planck's constant divided by mass of the proton times the wavelength of 325 times 10 to the minus 9 meters which is the speed of 1.22 meters per second.