GR0177 #33



Alternate Solutions 
danielkwalsh 20100905 18:25:18  A quick trick: the equation is correct, but for an extremely relativistic particle, as we clearly have here, the decay time in the lab frame can be well approximated as , since . This simplifies the math a lot more in solving for . This gives , which is closest to choice (D).   Herminso 20090921 13:42:50  Use . In the own rest frame of the pion , and thus:
.
But in the Lab frame,
.
Now, using the invariant ,
Thus, .
ramparts 20091007 13:32:13 
By far the best solution, because it's the most elegant and because it makes the algebra significantly easier without a calculator.

 

Comments 
danielkwalsh 20100905 18:25:18  A quick trick: the equation is correct, but for an extremely relativistic particle, as we clearly have here, the decay time in the lab frame can be well approximated as , since . This simplifies the math a lot more in solving for . This gives , which is closest to choice (D).   Herminso 20090921 13:42:50  Use . In the own rest frame of the pion , and thus:
.
But in the Lab frame,
.
Now, using the invariant ,
Thus, .
hanin 20091004 22:13:57 
Why ?

kroner 20091005 11:36:41 
The pion's own rest frame is the frame where it doesn't move so .

ramparts 20091007 13:32:13 
By far the best solution, because it's the most elegant and because it makes the algebra significantly easier without a calculator.

  Ning Bao 20080201 06:26:28  Pions don't reach speed of light (incidentally, this eliminates E). It must be going very close, however, to contract 30m to a little less than 3 meters to not violate this. This means D.
noether 20091105 15:28:05 
If 2.99*10^8 were listed as an answer, would you have chosen that?

gravity 20101110 00:37:22 
Yeah. This one was tricky. I did most of what everybody else did and ended with 30 c/(909) which I figured looked more like more like c than it did 2.98 x 10^8.
Gah. I should have known better! At least it's not the real test.

  StrangeQuark 20070512 12:03:43  To make this faster (avoid the "messy" fraction)
Note to start that this is not a photon thus answer E is out.
now as above,
L=v t
L =v t
some simplification steps
=
now plug and chug...
=
Note now that we have a
in the first term in the denominator, leaving only
in the denominator,
but
so we simplify
<
after canceling we see that
v<
but only by a very small amount thus we have D   boundforthefloor 20061126 05:25:22  Can anyone clarify this? I'm befuddled and can't find much that helps.
johnyp03 20061129 17:48:43 
So, you know L=30=vt in the lab frame. But, in the pion's frame, there is time dilation. So, t=t'=(gamma)t0 where t0=3*10^8, gamma=1/(sqrt(1v^2/c^2)). So:
30 = v(gamma)t0 = v*(10^8)/(sqrt(1v^2/c^2))
30*sqrt(1v^2/c^2) = v*(10^8)
900*(1v^2/c^2) = v^2(10^16)
900 = v^2*10^16 + v^2*900/c^2
==================> 900/c^2=(900/9)*10^16=10^14
900 = v^2(10^16 + 10^14)
sqrt(900/(10^16+10^14)) = v
30/sqrt(1.01*10^14) = v
30/1.005*10^7 = v
v ~ 2.98*10^8
Hope this helps

boundforthefloor 20061201 11:40:08 
Thanks johnyp. The actual equation setup was killing me, now that I've seen the calcualtions it makes much more sense.

VanishingHitchwriter 20061201 14:07:02 
Surround your expressions with dollar signs for latex equations. Here's the guy's comment back again...
where , . So:
==================>

matweiss 20100929 09:31:49 
hey, I don't understand why t0 is 3 x 10^8. it seems like I thought t in the frame of the particle was 10^8 s and that (if you assume the particle to essentially be moving at light speed) then t in the lab frame equals 30/c= 1x 10^7. any help?

  senatez 20061102 12:46:13  I ended up with a messy fraction 10c/10.1 which you estimate as .99c. This is about 2.98*10^8. I sure wish they would give easier arithmatic, it is waists a lot of time.  

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