Relativistic velocity transformations

[trevorr93]

Homework Statement

An excited nucleus of krypton-80 emits a gamma ray that travels at the speed of light relative to the nucleus. The nucleus itself has a speed of 0.60c relative to the sun. Use a relativistic velocity transformation to determine the speed of the gamma ray relative to the sun if the gamma ray is emitted: a) in the direction of motion of the nucleus and b) opposite to the direction of motion of the nucleus

The Attempt at a Solution

a)

u = u' + v / 1 + vu'/c^2
u = 1 + 0.6 / 1 +(0.6)(1)/c^2
u = 1 c

Does this make sense? Can a particle not move faster then the speed of light?

b) Not sure how to approach this! would a number be negative?

 

[CompuChip]

Looks like you fell for it :-)

A gamma ray is actually just light. That's why it can, and in fact must, travel at the speed of light. What you have shown in a) is one of the postulates of relativity (from which the transformation rule for velocities is derived), namely that light travels at the speed of light for all observers, irrespective of their relative velocity.

Now before you do b), try to think what the answer should be ... then plug in the numbers and see if you are right. If you did get a negative number, how would you explain that?

 

[jayjay713]

oh okay. so b would be equal to 1 as well because of the postulate. thank you for your help!

 

[CompuChip]

Yep*, but you should also be able to get this explicitly from the formula!

*) Actually, almost yep! - I suggest to give it a try anyway.

 

[asteriatic]

I would like to first clarify that the concept of relativistic velocity transformations is a fundamental aspect of Einstein's theory of relativity, which describes the relationship between space and time. In this theory, the speed of light is considered to be a constant and the maximum speed at which any object can travel. Therefore, the statement "Can a particle not move faster than the speed of light?" is correct and is a fundamental principle in physics.

Now, to answer the question at hand, let's first define some variables. In this scenario, u represents the speed of the gamma ray relative to the sun, u' represents the speed of the gamma ray relative to the nucleus, and v represents the speed of the nucleus relative to the sun.

a) Using the relativistic velocity transformation formula, we can determine the speed of the gamma ray relative to the sun if it is emitted in the direction of motion of the nucleus. Plugging in the given values, we get:

u = (0.60c + c) / (1 + (0.60c)(c)/c^2)
u = 1.20c / (1 + 0.60)
u = 0.48c

This means that the gamma ray will be moving at 0.48 times the speed of light relative to the sun.

b) To determine the speed of the gamma ray relative to the sun if it is emitted in the opposite direction of the nucleus, we simply change the sign of the velocity of the nucleus in the formula. This is because the direction of motion of the nucleus is opposite to the direction of the gamma ray. Plugging in the values, we get:

u = (0.60c - c) / (1 + (0.60c)(-c)/c^2)
u = -0.40c / (1 - 0.60)
u = -0.67c

This means that the gamma ray will be moving at 0.67 times the speed of light in the opposite direction of the nucleus relative to the sun.

In conclusion, the relativistic velocity transformations allow us to accurately determine the speed of an object relative to another object, taking into account the effects of relativity. It is important to note that although the speed of light is considered to be a constant, the relative speeds of objects can still vary and can even exceed the speed of light in certain situations, such as in the case of