# Energy of proton moving fast (special relativity)

In summary, the problem involves a proton crossing the Milky Way in 5 minutes in its own reference frame. Using the equations for special relativity, we can calculate the approximate energy of the proton and the time it would take for the proton to cross the galaxy as measured by an observer in the galaxy's reference frame. However, there may be an error in assuming the galaxy is 10^5 meters wide from the point of view of the proton, and further calculations may be necessary to accurately determine the velocity and solve the problem.

## Homework Statement

Given: in its own reference frame a proton takes 5 minutes to cross the Milky Way (10^5 meters).
(a) What is the approximate energy of the proton?
(b) About how long would the proton take to cross the galaxy as measured by an observer in the galaxy's reference frame?

## Homework Equations

All those equations for special relativity I imagine.
L = L0 Sqrt[1-(v/c)^2]
t = t0 Sqrt[1-(v/c)^2]

## The Attempt at a Solution

So I keep doing this over and over again, getting a wrong answer. The way I approach it is this: I think L is the distance as measured by the proton when crossing the galaxy, and t is the time the proton measures (5 minutes). Since v is the same in both reference frames (how fast the galaxy thinks the proton is moving and how fast the proton thinks the galaxy is moving) I can say that L/t = v. Using those equations I do it all out and find that v = c. Not very useful when I plug it into my equations for energy and find the proton has infinite energy. Where am I going wrong?

Also, I think this should be doable without the Lorentz transformations.

I'm not very confident about relativity, but I think you may have an error in assuming the galaxy is 10^5 meters wide from the point of view of the proton. I expect you have to use one of the formulas to transform this to the proton's point of view. Or else transform the 5 minutes to the galaxy's point of view. Before calculating the velocity. Of course you can't do that numerically, so you'll have an expression with a v in it rather than a number. Hopefully after the next step - find v - you will be able to get the two v's together somehow and solve for v.

Since t0 is the time measured by the proton, I can just use that to find the velocity (v = L/t0). Then use that to find the energy. But I still get an infinite answer.I would approach this problem by first clarifying the given information and assumptions. The statement says that the proton takes 5 minutes to cross the Milky Way in its own reference frame. This implies that the proton is moving at a constant speed in its own reference frame. However, in special relativity, the speed of an object is relative and can be different in different reference frames. So, is the given time based on the speed of the proton in its own reference frame or in the reference frame of the galaxy?

Assuming that the given time is based on the speed of the proton in its own reference frame, we can use the Lorentz transformations to find the speed of the proton in the reference frame of the galaxy. Using the equation t = t0 Sqrt[1-(v/c)^2], we can solve for v to find that the speed of the proton in the reference frame of the galaxy is v = 0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

## 1. What is the concept of special relativity?

Special relativity is a theory developed by Albert Einstein that describes the relationship between time and space when objects are moving at high speeds, close to the speed of light.

## 2. How does special relativity affect the energy of a moving proton?

According to special relativity, the energy of a moving proton increases as its velocity increases. This effect is known as relativistic mass increase.

## 3. What is the formula for calculating the energy of a moving proton in special relativity?

The formula for calculating the energy of a moving proton in special relativity is E = mc^2 / √(1 - v^2/c^2), where E is energy, m is mass, c is the speed of light, and v is the velocity of the proton.

## 4. How does the energy of a moving proton compare to its rest energy in special relativity?

In special relativity, the energy of a moving proton is always greater than its rest energy. As the velocity of the proton approaches the speed of light, its energy approaches infinite.

## 5. Can the energy of a moving proton ever exceed the speed of light in special relativity?

No, according to special relativity, the speed of light is the maximum speed possible for any object. Therefore, the energy of a moving proton can never exceed the speed of light.

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