Relativistic Velocity Addition

In summary: Use the relativistic velocity addition formula to find v, the velocity of A relative to Europa.In summary, the problem involves two asteroids moving towards each other at the same speed as measured from Europa. Their relative speed is 0.5c and you are asked to find the speed of one of the asteroids relative to Europa. Using the relativistic velocity addition formula, you can find this speed by considering the velocities of the two asteroids in the rest-frame of Europa. There is no ambiguity in the problem and you can solve it by setting up and solving a simple equation.
  • #1
mfreeman
3
0

Homework Statement


Two asteroids are approaching one another moving with the same speed speed as measured from a stationary observer on Europa. Their relative speed is 0.5c. Find the speed of one of asteroids relative to Europa.

I understand how relativistic velocity addition works but am not able to solve this question. Is it possible there is not enough information given in the problem? Any help would be much appreciated.

Homework Equations


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

The Attempt at a Solution


The ambiguity of the problem statement has led me down several different paths each resulting in a quadratic that in most cases results in a speed faster than c.
 
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  • #2
You might find the following form of the equation easier to understand.

Relativistic addition of velocities:
[tex]V_{a/c} = \frac{V_{a/b} + V_{b/c}}{1 + (V_{a/b} V_{b/c})/c^2}[/tex]

Hint: Let "b" be Europa. ("a/c" means the velocity of "a" as measured in the frame of "c".)
 
  • #3
Assuming "b" is Europa I would let V(a/c) be the speed of Ship A relative to Ship B which we know to be 0.50c. By the problem statement we have V(a/b) = V(b/c) = V(c/b) = V'. Thus we have 0.50c = 2V' / (1 + (V'^2/c^2)). This leads to a quadratic which I was told is the incorrect approach. Was that incorrect advice? Thank you very much for your help.
 
  • #4
mfreeman said:
Assuming "b" is Europa I would let V(a/c) be the speed of Ship A relative to Ship B which we know to be 0.50c.
To keep your sanity, let "a" stand for asteroid #1 and "c" stand for asteroid #2. Furthermore, let asteroid "a" move to the right and "c" move to the left. (As seen from Europa.)

mfreeman said:
By the problem statement we have V(a/b) = V(b/c) = V(c/b) = V'.
Careful! Signs matter. Let "to the right" be positive.
 
  • #5
mfreeman said:
Thus we have 0.50c = 2V' / (1 + (V'^2/c^2)).
Actually, this equation looks fine to me.
 
  • #6
Gosh I thought so! Thanks.
 
  • #7
mfreeman said:
Gosh I thought so! Thanks.
Good!

Just for the record, if V(a/b) = V', then V(b/a) = - V'.
 
  • #8
mfreeman said:

Homework Statement


Two asteroids are approaching one another moving with the same speed speed as measured from a stationary observer on Europa. Their relative speed is 0.5c. Find the speed of one of asteroids relative to Europa.

I understand how relativistic velocity addition works but am not able to solve this question. Is it possible there is not enough information given in the problem? Any help would be much appreciated.

Homework Equations


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

The Attempt at a Solution


The ambiguity of the problem statement has led me down several different paths each resulting in a quadratic that in most cases results in a speed faster than c.

There is no ambiguity. Relative to Europa, asteroid A moves at velocity +v and asteroid B at velocity -v. You are given that the velocity of B in the rest-frame of A is -c/2.
 

Related to Relativistic Velocity Addition

1. What is Relativistic Velocity Addition?

Relativistic Velocity Addition is a mathematical formula used to calculate the combined velocity of two objects moving at relativistic speeds.

2. Why is Relativistic Velocity Addition important?

Relativistic Velocity Addition is important because it allows us to accurately calculate the velocity of objects moving at speeds close to the speed of light, which is crucial in fields such as astrophysics and particle physics.

3. How does Relativistic Velocity Addition differ from classical velocity addition?

Classical velocity addition, also known as Galilean velocity addition, follows a linear formula where velocities are simply added together. Relativistic Velocity Addition takes into account the effects of time dilation and length contraction at high speeds, resulting in a non-linear formula.

4. Can Relativistic Velocity Addition be used for objects moving in opposite directions?

Yes, Relativistic Velocity Addition can be used for objects moving in opposite directions. In this case, the resulting velocity will have a negative sign, indicating the direction of motion.

5. What are some real-world applications of Relativistic Velocity Addition?

Relativistic Velocity Addition is used in various fields such as astronomy, particle accelerators, and space travel. It is also important in understanding the behavior of high-energy particles and their collisions.

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