WEBVTT
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This is College Physics Answers
with Shaun Dychko.
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A Supernova occurred
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in the Magellanic Cloud in 1987
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and that cloud is 120,000 light years away
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and some neutrinos were detected on Earth
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as well as light photons, you know,
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light from the Supernova was
also detected
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but there's a time difference between
when they were detected.
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The neutrinos were slightly slower
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and that's on account of their mass;
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to the extent that they have mass,
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they are not capable of traveling
at the speed of light.
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So we are going to estimate based on
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kinetic energy of 700 kiloelectron volts
for the neutrinos
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and an estimated mass of 7 electron volts
per *c squared*,
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what is the time difference between
receiving the photons
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versus receiving the neutrinos?
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The question in part (a) asks
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what is this Lorentz factor *γ*?
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So we know that kinetic energy is
*γ* minus 1
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times rest mass times *c squared*
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and then we can expand the bracket by
multiplying through
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by *mc squared here* and we have
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*γmc squared* minus *mc squared*,
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we'll add *mc squared* to both sides
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and we have *γ* times *mc squared* equals
kinetic energy plus *mc squared*
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then divide both sides by *mc squared*
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and then we get *γ* is kinetic energy plus
*mc squared* all over *mc squared*
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and that's 700 times 10 to
the 3 electron volts
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plus 7 electron volts per *c squared*
times *c squared*
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divided by 7 electron volts per *c squared*
times *c squared*
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and this works out to 100001—
that's the value of gamma.
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In part (b), we have to find
this time difference.
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Well, let's first figure out
what is the distance
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and this is a distance of 120000 light years,
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which means it's the distance that
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light would travel in 1 year times 120000.
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So this *l* represents the speed of light
and this *y* represents years
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so sometimes, I like to abbreviate this
as *c.y*—
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the speed of light multiplied by
a time period of 1 year.
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So we are going to turn this into
units of *c* times *s*
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and this is
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also a bit strange because, you know,
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normally, I would convert this
into a distance of meters
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but I am not going to do that here
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because I want this speed of light...
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I want this distance as a factor of
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or a factor times the speed of light
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because the speed of light was going to
cancel conveniently later on.
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So we are just going to convert
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the year unit though into seconds
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so we multiply by 365.25 days
per year and then
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24 hours per day and then
3600 seconds per hour
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and we are left with 3.786912 times
10 to the 12
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times the speed of light-seconds.
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Alright!
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So if that seems strange
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well, it's just as strange as light years
and this is now just light seconds.
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Okay!
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The time for the neutrino to travel
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is going to be this distance divided by
the speed of the neutrino
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and we need to
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solve for *v* in this expression for
the Lorentz factor gamma.
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So we have...
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we can square both sides
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and so we have *γ squared* is 1 over
1 minus *v squared* over *c squared*
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then multiply both sides by 1 minus
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*v squared* over *c squared*
divided by *γ squared*
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and then we get 1 minus *v squared*
over *c squared* equals
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1 over *γ squared*
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and then we'll add *v squared* over
*c squared* to both sides
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and subtract 1 over *γ squared*
from both sides
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then switch the sides around
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and then we get *v squared* over
*c squared* equals
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1 minus 1 over *γ squared*,
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multiply both sides by *c squared*
and you get this line,
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take the square root of both sides
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and you have *v* then is the speed of
light times square root
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1 minus 1 over *γ squared*.
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And I am not going to substitute in
a number for *c*,
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I am just going to leave it
as the factor *c*
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and for the same reason that
I left this factor *c*
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in this distance as well,
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it's going to cancel away later on.
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So we have the square root of 1
minus 1 over 100001 squared
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is this number and this number comes from
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spreadsheet program Excel
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because the calculator isn't
capable of handling
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so many digits but
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but Excel spreadsheet can
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so this is square root 1 minus
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1 over 100001 squared
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and that makes 0.9 with ten 9s
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a 5 and then 0001
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so it's that factor times *c* is the speed.
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Okay!
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Now the time difference then
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is going to be the time it takes for
the neutrino to arrive
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minus the time it takes
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for a photon to arrive traveling
at the speed of light *c*.
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So that's the distance divided by
the speed of the neutrino minus
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the distance divided by the speed of light
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and we can factor the *d* out and
we have *d* times 1 over
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speed of the neutrino minus
1 over speed of light.
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Now I don't want to write in
the number for
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the speed of light because
we are dealing with
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such small...
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we are dealing with numbers that
are so precise
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I would be introducing a problem by
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entering in a speed for the speed of light
unless I had
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you know, this many digits in
the speed of light
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which I don't feel like writing down
because that's just
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inconvenient to write down
so many digits
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and I did it here but I don't want to
do it for the speed of light as well
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and there's a way to get around that issue
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by multiplying this by
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the number 1
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but number 1 is going to look funny—
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it's going to look like *c* over *c*
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and then leave the *c* in the denominator
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for this fraction and then multiply
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the top *c* into the brackets.
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So we have *d* over *c*
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times *c* over the speed of the neutrino
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*c* over *c*, which is 1.
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And this is convenient to have
this expression because
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we have *c* divided by some factor
multiplied by *c*
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and so the *c*'s cancel
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and then this distance is some factor
multiplied by *c*
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which cancels with this *c* here
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and we are left with seconds
in our units here
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times 1 over this big number
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well, it's a small number but, you know,
many digits... anyhow...
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minus 1.
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Now this again is something that
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a calculator brain can't really handle
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so we turn to the spreadsheet
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and it's going to be
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this number here, this time
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it was a distance but then
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the *c* canceled with the *c* in
the denominator here
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so it's 3.786912 times 10 to
the 12 seconds—
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that's this part here—
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times 1 over this denominator here
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(whoops)
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which is the cell A1 minus 1.
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So I have plugged this into the Excel
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and it gives the answer 189 so
that's 189 seconds.
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So there's 189 seconds between when
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the photons are detected from
the Supernova
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and the neutrinos are then detected
189 seconds later.