# Solve Relativity Problem: Proton from Moon to Earth

• dpeagler
In summary, a cosmic ray proton with a mass of 1836.15267261(85) times that of an electron is moving at a speed of 0.8c from the moon to the earth, a distance of 3.844*10^5 km. It takes a certain number of seconds for the proton to travel this distance, and its rest energy in MeV is known. The proton's kinetic energy and total energy in MeV as seen from earth can be calculated, as well as the value of $sqrt[E^2-(pc)^2]$ in the Earth's frame. The velocity of the proton is relative to the Earth, and is 0.8c.
dpeagler

## Homework Statement

A cosmic ray proton is moving at a speed of 0.8c as it travels from the moon to the earth. The distance from the moon to the Earth is 3.844*10^5 km (ignore any motions of the Earth and moon). The mass of an electon is 0.511 MeV/c^2 and the proton to electron mass ratio is 1836.15267261(85). Let the proton be in S' and the Earth in frame S.

a. How many seconds does it take for this proton to travel from the moon to the earth?
b. What is the rest energy in MeV of this proton
c. What is the kinetic energy, in MeV, of this proton as seen from earth?
d. What is the total energy, in MeV, of this proton as seen from earth?
e. What is the value of $sqrt[E^2-(pc)^2]$ in the Earth's frame for this proton

## The Attempt at a Solution

I know how to get b., but am confused about the velocity. Is the velocity relative to Earth or for the proton. I know I can check my answers by solving e. b/c if I am thinking correctly that expresssion in part e. should always be equal to the rest energy mc^2.

If someone could just help me figure out what velocities I'm supposed to be working with I could figure it out. But right now I am just getting confused when I plug the stuff into the momentum equation and kinetic energy equation and so forth.

Thanks for any help.

Relative to the Earth, the proton's speed is 0.8c.
Relative to itself, the proton's speed is zero.

## 1. How does relativity affect the travel of a proton from the Moon to Earth?

Relativity plays a significant role in the travel of a proton from the Moon to Earth. According to Einstein's theory of relativity, time and space are relative and can be affected by the speed and gravity of objects. This means that the travel time of a proton from the Moon to Earth will be affected by the speed at which it travels and the gravitational forces exerted by both celestial bodies.

## 2. What is the speed of light and why is it important in solving this problem?

The speed of light, denoted by the symbol 'c', is approximately 299,792,458 meters per second. It is considered to be the maximum speed at which all matter and information can travel. In the context of this problem, the speed of light is important because it is the speed limit that a proton can travel at. Therefore, it is crucial to consider the speed of light when calculating the travel time of a proton from the Moon to Earth.

## 3. How can the distance between the Moon and Earth impact the travel time of a proton?

The distance between the Moon and Earth can greatly impact the travel time of a proton. As the distance between two objects increases, the gravitational force between them decreases. Therefore, the proton will experience a weaker gravitational pull from Earth as it travels from the Moon. This can result in a longer travel time as the proton will not be able to accelerate as much towards Earth.

## 4. What factors must be considered in solving this relativity problem?

There are several factors that must be considered in solving this relativity problem. These include the speed of light, the distance between the Moon and Earth, the gravitational forces exerted by both celestial bodies, and the speed at which the proton is traveling. Additionally, the effects of time dilation and length contraction due to relativity must also be taken into account.

## 5. Can the theory of relativity be applied to other scenarios besides the travel of a proton from the Moon to Earth?

Yes, the theory of relativity can be applied to various other scenarios besides the travel of a proton from the Moon to Earth. It is a fundamental theory that explains the relationship between space, time, and gravity. It has been applied in many areas of science, including astrophysics, cosmology, and particle physics. Additionally, everyday scenarios such as GPS systems and satellite communications also rely on the principles of relativity to function accurately.

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