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(a) Verify from its quark composition that the $\Delta^+$ particle could be an excited state of the proton. (b) There is a spread of about 100 MeV in the decay energy of the $\Delta^+$, interpreted as uncertainty due to its short lifetime. What is its approximate lifetime? (c) Does its decay proceed via the strong or weak force? Note: at 2:50 I misspoke saying 10 to the minus "six", when it should have been 10 to the minus "sixteen".
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  2. $3.3\times 10^{-24}\textrm{ s}$
  3. Since quark flavor is conserved this decay is probably mediated by the strong nuclear force. The short lifetime of $\approx 10^{-24}\textrm{ s}$ also indicates the strong nuclear force.
Solution Video

OpenStax College Physics for AP® Courses Solution, Chapter 33, Problem 21 (Problems & Exercises) (3:01)

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Video Transcript
This is College Physics Answers with Shaun Dychko. The delta plus particle has the same quark composition as a proton; it's up, up, down, those are the flavors of the quarks in the delta plus particle and you can find this from table [33.4]. In part (b) it says, the uncertainty in rhe energy of the delta plus particle is about 100 megaelectron volts and the question then is what is its approximate lifetime given this uncertainty in its energy? Well the Heisenberg uncertainty principle says that product of the uncertainty in energy and time has to be greater than or equal to Planck's constant divided by 4π and so we'll solve for the uncertainty in time and we'll take this to be a minimum possible lifetime. So we'll divide both sides by ΔE and we get the uncertainty in time then is Planck's constant over 4π times uncertainty in energy. So that's Planck's constant there divided by 4π times the energy written as times 10 to the 6 electron volts and then convert it into joules in order to have units that work together mks units and we are left with an uncertainty in time of 3.3 times 10 to the minus 24 seconds. And so this is the minimum possible lifetime that's measurable or it could be some number a bit bigger than that this is the uncertainty anyhow. OK And then the next question is is this decay of the delta plus into a proton plus the neutral pion is that decay mediated by the strong nuclear force or the weak nuclear force? Now if we look at the quark compositions, we have up, up, down in the case of the delta plus and we also have the same up, up, down in the case of the proton and then for this neutral pion, it's a mixture of up and then anti-up or down and anti-down and so there's no net change in the quark flavor for this process because you know this down stays a down, this up stays an up and this up stays an up and then the baryon number is conserved because for every up quark that appears anew, we have the anti-up also compensating for that. And then likewise for down and anti-down; those two make a net baryon number of zero. And so there's no change in the quark flavor and for that reason, we suspect that this is mediated by the strong force because a strong nuclear force is not able to change quark flavor. And the short lifetime of 10 to the minus 24 seconds also indicates the strong nuclear force; the strong nuclear force is involved in fast decays with short lifetimes typically the strong force has a lifetime of 10 to the minus 16 to 10 to the minus 23 seconds whereas the weak nuclear force will have lifetimes of 10 to the minus 12 to 10 to the minus 16 seconds.