Calculate the protons kinetic energy

[asdf1]

for the following question:
a proton with the total energy E=3E0please calculat the proton's (a) kinetic energy (b) velocity (c) momentum (d)mass

my problem:
1) can someone double check my work?
(a)Ek=2EE0
(c) p=[8^(0.5)]EE0/c
(d) I've tried
(mc^2)^2 + Ek^2 +2m(c^2)Ek=(mc^2)^2+(pc)^2
=>9E0^2+(2E03E0)=8E0^2
which is weird...
the rest I'm not sure how to calculate?

 

[quasar987]

But the problem doesn't even state that the proton is moving. This is absurd.

What formula did you use for (a). I can't make sense of it. Also, you say Ek = 2EE0. But E = 3E0. So we would have Ek = 6E0², right?

 

[µ³]

The proton must be moving by the sheer fact that the energy is greater than the rest energy (and we are not accounting for any other type of energies).
remember that
$$K = (\gamma-1) m_0 c^2=\gamma m_0 c^2 - m_0 c^2$$
but $m_0 c^2$ is the rest energy and $\gamma m_0 c^2$ is total energy
$$K= 3E_0 - E_0 = 2E_0$$
for d) Rest energy is
$$E_0 = m_0 c^2$$
so solve for m_0

 

[quasar987]

Oh, I didn't get that E0 denotes the rest energy...

 

[jtbell]

(a)Ek=2EE0
(c) p=[8^(0.5)]EE0/c

Neither of the above answers is correct, as you can tell by checking the units. In (a) you have units of energy on the left, but units of energy^2 on the right. That's a good sign that you omitted a square root somewhere. In (b) if you write your answer as $pc=\sqrt{8} E E_0$ you can see that you have the same trouble because pc has units of energy.

For part (d) you're supposed to find the mass. Which mass? Rest mass or relativistic mass? Either way, it's trivial. Hint: rest mass is to rest energy as relativistic mass is to (total) energy.

 

[asdf1]

so
(a)2Eo
(c)p=[(8)^0.5]Eo/c
(d) p^2/2m=[(8)^0.5]Eo/c => m=2Eo/c^2 but why doesn't this equal the rest mass m=E0/c^2?

 

[El Hombre Invisible]

total energy E=3E0

calculate the proton's (a) kinetic energy (b) velocity (c) momentum (d)mass

Backwards seems easiest. Assuming E0 is rest energy, then:

(d) If by mass, rest mass is meant, then m0 = E0/c^2.
If relativistic mass is meant, then m = 3E0/c^2.

(c) pc = ROOT(E^2 - m0^2c^4)
So p = ROOT(9E0^2/c^2 - E0^2/c^2)
= ROOT(8)E0/c

(b) v = pc^2/E
= ROOT(8)/3 x c
= 0.94c

(a) Ekin = E - E0 = 2E0

 

[asdf1]

why is the way that i calculated (d) incorrect?
also, where does "v = pc^2/E" come from in (b)?

 

[jtbell]

so
(a)2Eo
(c)p=[(8)^0.5]Eo/c
(d) p^2/2m=[(8)^0.5]Eo/c => m=2Eo/c^2 but why doesn't this equal the rest mass m=E0/c^2?

$p^2/2m$ is the non-relativistic kinetic energy, so your starting point for (d) says that the non-relativistic kinetic energy equals the relativistic momentum. How did you come up with that?

 

[El Hombre Invisible]

why is the way that i calculated (d) incorrect?

Couldn't follow your logic, so I don't know, but this might help:

If relativistic mass is required:

m = p/v
= (ROOT(8)E0/c) / (ROOT(8)/3 x c)
= 3E0/c^2

If rest mass is required, then:

p = m0v/ROOT(1 - v^2/c^2), so...
m^2 = p^2/v^2 * (1 - v^2/c^2)
= (8E0^2/c^2)/(8c^2/9)*(1 - (8c^2/9)/c^2)
= (9E0^2/c^4)*(1/9)
= E0^2/c^4, so...
m = E0/c^2

also, where does "v = pc^2/E" come from in (b)?

v = pc^2/E comes from:

p = mv, so...
v = p/m

m = E/c^2, so...
v = pc^2/m

 

[asdf1]

thank you very much! :)