Solving for c: Speed of Q2 in Q1 Reference Frame

[Pochen Liu]

Homework Statement

Stellar system Q1 moves away from us at a speed of 0.860c. Stellar system Q2, which lies in the same direction in space but is closer to us, is moving away from us at a speed 0.560c. What multiple of c gives the speed of Q2 as measured by an observer in the reference frame of Q1?

Relevant Equations

*attached

I've tried using this equation:

Where:
u' = 0.86c
v = 0.3c
u = Is the speed of Q2 as measured by an observer in the reference frame of Q1 = 0.922c

Where have I gone wrong? Have I missed a negative symbol?

 

[stevendaryl]

Before plugging the numbers in, make sure you understand the variables in velocity addition formula.

If you have an observer $A$, a second observer $B$, and an object $C$, then let:

• $v_{BA}$ = the velocity of $B$ as measured by $A$.
• $v_{CB}$ = the velocity of $C$ as measured by $B$.
• $v_{CA}$ = the velocity of $C$ as measured by $A$.

If all three velocities are in a straight line, then according to relativity,

$v_{CA} = \dfrac{v_{BA} + v_{CB}}{1+ \frac{v_{BA} v_{CB}}{c^2}}$

In your case, the three objects are:

1. You = $A$
2. Q1 = $B$
3. Q2 = $C$

In this case, what they are asking is not $v_{AC}$. They tell you that that is equal to $0.860c$. They are asking for $v_{BC}$.

So you have two options, that work the same:

• Write down the formula for $v_{CA}, v_{BA}, v_{CB}$ and solve, algebraically for $v_{CB}$ in terms of the other two, or
• Switch the labels around. Let $A$ be Q1, $B$ be you, and $C$ be Q2.

If you do the second one, you have to remember that if Q1 is moving at 0.860c relative to you, then you are moving at -0.850c relative to Q1.

 

[PeroK]

Problem Statement: Stellar system Q1 moves away from us at a speed of 0.860c. Stellar system Q2, which lies in the same direction in space but is closer to us, is moving away from us at a speed 0.560c. What multiple of c gives the speed of Q2 as measured by an observer in the reference frame of Q1?
Relevant Equations: *attached

I've tried using this equation:

View attachment 243343
Where:
u' = 0.86c
v = 0.3c
u = Is the speed of Q2 as measured by an observer in the reference frame of Q1 = 0.922c

Where have I gone wrong? Have I missed a negative symbol?

Try the calculation using the classical velocity addition and see what you get.