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GR0177 #79
Problem
 GREPhysics.NET Official Solution Alternate Solutions
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Special Relativity$\Rightarrow$}Rest Mass

The total relativistic energy is $10GeV=\gamma mc^2=\sqrt{p^2c^2+m^2c^4}$.
The total relativistic momentum is $p=8GeV/c = \gamma mv$.

Plugging the momentum into the first equation, one has $(8^2+m^2c^2)c^2=10^2 \Rightarrow m^2c^4 = 100-64 = 36 \Rightarrow m = 6GeV/c^2$, as in choice (D).

Alternate Solutions
 Deafro2012-01-24 15:45:45 E^2=sqrt(p^2*c^2+(m*c^2)^2) E=10 GeV p=8 GeV/c mc^2= rest energy Rearrange the formula and solve for mc^2.Reply to this comment mike2009-11-06 18:34:27 Another way of doing it: $\gamma m v = 8 /c$ and $\gamma m c^2 = 10$. Multiply the first eqn by $c^2$ and divide the two to get: $v = \frac{4}{5}c$ Plug in to $\gamma$ to find $\gamma = \frac{5}{3}$. Plug back in to the momentum equation: $\frac{5}{3} m \frac{4}{5} c = 8 / c$ thus $m = 6$. I omit units, but it should be easy enough to follow. It seems a little longer, but its much less thinking once one remembers the relativistic energy/momenta equations. Reply to this comment Ning Bao2008-02-01 07:43:40 Set c=1 -> pythagorean triple.Reply to this comment
Deafro
2012-01-24 15:45:45
E^2=sqrt(p^2*c^2+(m*c^2)^2)

E=10 GeV
p=8 GeV/c
mc^2= rest energy

Rearrange the formula and solve for mc^2.
mike
2009-11-06 18:34:27
Another way of doing it:

$\gamma m v = 8 /c$

and

$\gamma m c^2 = 10$. Multiply the first eqn by $c^2$ and divide the two to get:

$v = \frac{4}{5}c$

Plug in to $\gamma$ to find $\gamma = \frac{5}{3}$.

Plug back in to the momentum equation:

$\frac{5}{3} m \frac{4}{5} c = 8 / c$

thus $m = 6$.

I omit units, but it should be easy enough to follow. It seems a little longer, but its much less thinking once one remembers the relativistic energy/momenta equations.
astrodoo
2008-09-17 01:38:23
manasi, the value of 64 (<- the square of momentum) has a dimension of $GeV^{2}/c^{2}$, so you can easily catch the yosun's solution if you consider the dimension of each terms.
Ning Bao
2008-02-01 07:43:40
Set c=1 -> pythagorean triple.
Gaffer
2007-10-24 08:42:58
There is a typo in the soln, which is why manasi is confused. Yosun used the $\c^2 in p^2 c^2$ to cancel the GeV/c in the momentum. But then she pulled it out again inthe next line.

No biggie.

It should go:

Plugging the momentum into the first equation, one has $\8^2 (c/c)^2 + m^2 c^4 = 10^2$
and then continue as she concludes
manasi
2007-09-28 12:05:48
hi, if i m not mistaken shudnt it be $m^2 c^4 = 100-64c^2$ ????

hope it appears rite m new to latex!!
hey thanks a lot yosun.. ur solutions are very helpful!! :)
 astrodoo2008-09-17 01:46:59 manasi, the value of 64 (<- the square of the momentum) has a dimension of $GeV^{2}/c^{2}$, so you can easily catch Yosun's solution in which $c^{2}$ term is disappeared if you look dimensions of each term.

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