Relative Velocity of Two Rockets and the Earth

In summary, according to the homework statement, A measures B to be travelling at 0.997c, while the Earth frame travels at 0.75c.
  • #1
rbn251
19
0

Homework Statement


Two rockets A and B are moving away from the Earth in opposite directions at 0.85c and -0.75c respectively.
How fast does A measure B to be travelling?

Now I have worked out v = -0.85-0.75/(1- -0.85*-0.75) = -0.997. This is correct.

Now I would like to work it out backwards to check my understanding so:
According to A:
B travels at 0.997c
The Earth frame travels at 0.75c

Q - How fast does the Earth measure B to be travelling?

Homework Equations


w=u-v/(1-uv)

The Attempt at a Solution



I expected the answer to be 0.85c but:
v=0.997-0.75/(1- 0.997*0.75) == 0.979 and not 0.85.

Thanks for any help
 
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  • #2
rbn251 said:

Homework Statement


Two rockets A and B are moving away from the Earth in opposite directions at 0.85c and -0.75c respectively.
How fast does A measure B to be travelling?

Now I have worked out v = -0.85-0.75/(1- -0.85*-0.75) = -0.997. This is correct.

Now I would like to work it out backwards to check my understanding so:
According to A:
B travels at 0.997c
The Earth frame travels at 0.75c

Q - How fast does the Earth measure B to be travelling?

Homework Equations


w=u-v/(1-uv)

The Attempt at a Solution



I expected the answer to be 0.85c but:
v=0.997-0.75/(1- 0.997*0.75) == 0.979 and not 0.85.

Thanks for any help
Are you sure?

Technically you're missing some brackets in that equation.
 
  • #3
As far are I know everything is correct 'mathematically'. You are right about the brackets but I have used the equation correctly in both cases as such.
 
  • #4
rbn251 said:

Homework Statement


Two rockets A and B are moving away from the Earth in opposite directions at 0.85c and -0.75c respectively.
How fast does A measure B to be travelling?

Now I have worked out v = -0.85-0.75/(1- -0.85*-0.75) = -0.997. This is correct.

Now I would like to work it out backwards to check my understanding so:
According to A:
B travels at 0.997c
The Earth frame travels at 0.75c

Your problem is just keeping your variables straight. In my opinion, it's always good to make up variables for all the values in a problem, and do as much of the derivation as possible symbolically, and only substitute numbers into the equations at the last step.

So, let's make up some variable names:
[itex]v_{AE} = [/itex] velocity of [itex]A[/itex] as measured in the Earth's rest frame
[itex]v_{BE} = [/itex] velocity of [itex]B[/itex] as measured in Earth's rest frame
[itex]v_{BA} = [/itex] velocity of [itex]B[/itex] as measured in A's rest frame

The velocity addition formula tells you that:

[itex]v_{BE} = \dfrac{v_{BA} + v_{AE}}{1 + \frac{v_{BA} v_{AE}}{c^2}}[/itex]

You have: [itex]v_{BA} = -0.997c[/itex], [itex]v_{AE} = 0.85c[/itex].

[edit]: [itex]v_{BA}[/itex] is negative, so it should be [itex]-0.997c[/itex].
[second edit]: As PeroK points out, it should actually be [itex]-0.977c[/itex]
 
  • #5
I think it should be 0.977 not 0.997. That's the problem.
 
  • #6
haha I think you're right - typo in the textbook!
 
  • #7
yup f*s sry for bothering!
 
  • #8
I prefer:

##v' = \frac{u+v}{1+uv}##

##v'' = \frac{v'-u}{1-v'u} = \frac{v(1-u^2)}{1-u^2} = v##

Then you can relax!
 
  • #9
PeroK said:
I prefer:

##v' = \frac{u+v}{1+uv}##

##v'' = \frac{v'-u}{1-v'u} = \frac{v(1-u^2)}{1-u^2} = v##

Then you can relax!

But my point is that for each of the variables [itex]u, v, v', v''[/itex] you need to get clear in your mind: Whose velocity is it, and relative to which frame? And you also have to keep in mind that velocity has a direction, as well as a magnitude. (In these 1-D problems, "direction" means "sign")
 

What is SR Velocity Transformation?

SR Velocity Transformation refers to the mathematical equations used in special relativity to transform velocities between two reference frames that are moving relative to each other at constant speeds.

Why is SR Velocity Transformation important?

SR Velocity Transformation is important because it allows us to accurately describe and predict how velocities will change between different reference frames in special relativity, which is essential for understanding the behavior of objects moving at high speeds.

How is SR Velocity Transformation calculated?

SR Velocity Transformation is calculated using the Lorentz transformation equations, which take into account the principles of time dilation and length contraction in special relativity.

What are the limitations of SR Velocity Transformation?

SR Velocity Transformation is only applicable to situations where objects are moving at constant speeds and in straight lines. It also does not take into account the effects of gravity or acceleration.

What are some real-world applications of SR Velocity Transformation?

SR Velocity Transformation has many practical applications in fields such as physics, engineering, and astronomy. It is used in the design of space missions, particle accelerators, and GPS systems, among others.

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