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In this problem I was solving a beam of uniform cross section carrying a current of $0.25\ \mu A$ by alpha particles. The mass of an alpha particle is $m_\alpha=4m_p$, where $m_p$ is the mass of a proton. Each alpha particle moves with a constant kinetic energy, $= 20\ MeV$.

If we calculate the speed of a particle, we get $v_{\alpha}= \sqrt{\frac{2\times20\times10^6\ eV}{4m_p}}=7.732\times10^{16}\ m/s$

How is this possible? Please explain.

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It can't. The fact that you're getting a speed that's larger than the speed of light is a sign something's gone wrong.

  • Check your units. You should not get something larger than $c$ (I just checked with a calculator).
  • For a different problem, if you do get something larger than $c$ (or even $>10\%$ of $c$, depending on how accurate you need your answer to be), then switch to the relativistic formula for kinetic energy - $KE = (\gamma-1) mc^2$.
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    $\begingroup$ For that matter, any result larger that 1%–10% of $c$ should be redone using relativistic physics. The exact cutoff depends on how much precision your answer needs to have. $\endgroup$
    – rob
    May 23 at 15:13
  • $\begingroup$ Good point, edited that in. $\endgroup$
    – Allure
    May 23 at 15:20
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Your calculations are incorrect, non-relativistic solution : $$v=\sqrt {\frac Em} = \sqrt \frac {20~MeV}{4 \times m_p} = \sqrt \frac {3.204 × 10^{−12} J}{4 \times 1.673 × 10^{−27} kg} = 2.188 × 10^7 m/s = 0.07c$$

For calculations like that, for veryfing your results, I would recommend a superb physics calculator Qalculate which can automatically convert to required units on the fly, i.e. you could write calculator code like

sqrt((20 × MeV)/(4 × proton_mass)) to c

Which is awesome !

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    $\begingroup$ Save some macroscopic conversions with $$\frac vc \approx \sqrt{\frac {2E}{mc^2}} \approx\sqrt{\frac{2\times 20\,\rm MeV}{4\,\rm GeV}}.$$ This approach also tells you, for free, that $\gamma-1$ is smallish. $\endgroup$
    – rob
    May 23 at 15:24
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    $\begingroup$ You lost the factor of 2 in the numerator. BTW, Google can do stuff like this, eg sqrt(2*(20 MeV)/(4*(proton mass))). We probably should use the relativistic equation here; OTOH, the error in using the Newtonian formula is less than 1%, which is less than the error due to that approximation for the alpha particle mass. $\endgroup$
    – PM 2Ring
    May 23 at 15:37
  • $\begingroup$ @PM2Ring Indeed, factor of 2 is lost, however due to sqrt() asymptotic behavior, that doesn't change answer very much. I'm sure Google does a lot of good stuff too, but calculator solves simple equations too and is optimized for working offline, and is free to. Highly recommend it! $\endgroup$ May 23 at 15:52

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