Proton passes the Milky Way galaxy, find its energy?

[asdf1]

for the following question:
In its own frame of reference, a proton takes 5min to cross the Milky Way galaxy, which is about 10^5 light-years in diameter.
What is the approximate energy of the proton in electronvolts?

my problem:
v*5*60=(10^5)*3*(10^8)*[1-(v^2/c^2)]
however, i can't calulate v on the calculator, because v is too small...
i was trying to use E=(1/2)mv^2, where m is the photon rest mass...
is that right?

@@a

 

[quantumdude]

v*5*60=(10^5)*3*(10^8)*[1-(v^2/c^2)]

This looks like algebraic soup! Why, oh why, are you combining these quantities together in this way? From what equation are you working?

It looks like:

$$vt=\frac{Dc}{\gamma ^2}$$

where $v$ is the proton speed in the galaxy frame, $D$ is the galactic radius in the galaxy frame, and $t$ is the travel time in the proton frame.

however, i can't calulate v on the calculator, because v is too small...
i was trying to use E=(1/2)mv^2, where m is the photon rest mass...
is that right?

No, it's not right. The photon rest mass is zero.

 

[quicknote]

I would approach this question by first clarifying some information. Is the proton traveling at a constant velocity or is it accelerating? Also, is the 5 minutes of travel time in the proton's frame of reference or in a stationary observer's frame of reference? These details will affect the calculation of the proton's energy.

Assuming that the proton is traveling at a constant velocity and the 5 minutes of travel time is in the proton's frame of reference, we can use the equation E = mc^2/[sqrt(1-(v^2/c^2))] to calculate the energy of the proton. In this equation, m is the rest mass of the proton, c is the speed of light, and v is the velocity of the proton.

Since we know the diameter of the Milky Way galaxy is approximately 10^5 light-years, we can calculate the velocity of the proton by dividing the distance by the time it takes to cross it. This gives us a velocity of approximately 1.26 x 10^13 m/s.

Plugging in this value for v, along with the rest mass of a proton (approximately 1.67 x 10^-27 kg) and the speed of light (3 x 10^8 m/s), we get an approximate energy of 1.5 x 10^20 electronvolts (eV) for the proton.

However, this calculation assumes that the proton is traveling at a constant velocity, which may not be the case. If the proton is accelerating, the calculation becomes more complex and we would need more information to accurately determine its energy. Additionally, if the 5 minutes of travel time is in a stationary observer's frame of reference, the calculation would also be different. It is important to consider all factors and clarify the details before making any conclusions about the energy of the proton.