Calculating Relative Velocity of Spaceships Using GRE Relativity Equations

[Silviu]

Homework Statement

Two spaceships pass each other. Space-ship A moves relative to a nearby planet at velocity $v_1$, while spaceship B moves at velocity $v_2$ relative to the planet. How fast does spaceship A move relative to spaceship B?

Homework Equations

The Attempt at a Solution

I just applied the addition of velocities $v = \frac{v_1+v_2}{1+v_1v_2/c^2}$. However the correct answer is $v = \frac{|v_1-v_2|}{1-v_1v_2/c^2}$. Can someone explain to me why? Also, if we take the classical case when 2 spaceships pass each other the relative velocity would be $v_1+v_2$, which is given by my solution. Thank you

 

[phyzguy]

Does your solution work in the classical case? Suppose A and B are both moving to the left at 10 miles/hour relative to the planet. Is their relative velocity 20 miles/hour?

 

[Silviu]

Does your solution work in the classical case? Suppose A and B are both moving to the left at 10 miles/hour relative to the planet. Is their relative velocity 20 miles/hour?

But the problem specifies that they are passing by each other. As we have no information about the acutal values of velocities, don't we have to assume they are moving toward each other (otherwise they might never meet)?

 

[phyzguy]

Since both velocities are specified as relative to the planet, I think you have to assume that they are specified in the same reference frame. Then, if they are moving toward each other, they must have opposite signs. I think your solution assumes both v1 and v2 are positive numbers.

 

[Silviu]

Since both velocities are specified as relative to the planet, I think you have to assume that they are specified in the same reference frame. Then, if they are moving toward each other, they must have opposite signs. I think your solution assumes both v1 and v2 are positive numbers.

I am confused.If they move towards each other, classically you have $v_1+v_2$. Why in this case you would have minus?

 

[phyzguy]

I am confused.If they move towards each other, classically you have $v_1+v_2$. Why in this case you would have minus?

If v1 and v2 are measured in the same reference frame, then their relative velocity of 1 relative to 2 is v1-v2. I think you are again implicitly assuming that v1 and v2 are both positive numbers. If they are measured in the same frame and they are moving toward each other, one is positive and the other negative. For what it is worth, I don't think the problem is very well worded. Usually we assume that a velocity is a vector quantity and we use the word speed for the magnitude of the velocity. Here the problem is mixing those two concepts.

 

[kuruman]

Here the problem is mixing those two concepts.

I am not so sure. Using the double subscript convention, let
$v_{AP}$ = velocity of spaceship A relative to Planet
$v_{BP}$ = velocity of spaceship B relative to Planet
$v_{BA}$ = velocity of spaceship B relative to spaceship A
Then
$$v_{BP}=\frac{v_{AP}+v_{BA}}{1+v_{AP}~v_{BA}/c^2}$$
which can be solved for $v_{BA}$ to get
$$v_{BA}=\frac{v_{BP}-v_{AP}}{1-v_{BP}~v_{AP}/c^2}$$
Note that the v's are velocities, not speeds, that could be positive or negative depending on what else is given. Also note that $v_{AB} = -v_{BA}$, hence the absolute value sign in the answer covers either case since the question asks

How fast does spaceship A move relative to spaceship B?