# Relative motion of two observers

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1. May 27, 2017

### Rahulrj

1. The problem statement, all variables and given/known data
Two observers having identical instruments carry out identical experiments to study the motion of a massive particle. First observer concludes that the particle is moving in a straight line with constant velocity while the second observer concludes that the particle is moving with acceleration. Which of the following statements made about the observers are correct:
a) both observers are at rest
b) second observer is moving with constant velocity w.r.t the first observer
c) both observers are moving with constant velocity w.r.t to a distant star
d) second observer is accelerating w.r.t first observer
2. Relevant equations
$r_{ab} = r_a - r_b$
3. The attempt at a solution
To my reasoning, both observers can't be at rest or have constant velocity in the same frame because according to relativity laws of physics are same in all inertial frames. However if I assume them to be in different frames of reference it is possible they can make that observation being at rest w.r.t their reference frame but here $a$ is a blanket statement that doesn't make much sense to me. My immediate reaction to this question was statement $d$ but I am not sure if it is correct because I am not able to properly reason why the other statements might be wrong. So I would like to know about the cases made in each statement.

2. May 27, 2017

### FactChecker

What is r in your relevant equations?
You are given information about velocities and accelerations. Do you have any relevant equations for velocity and acceleration?

3. May 27, 2017

### Rahulrj

Isn't it a conceptual question? I didn't think it requires equations to be solved. Anyway the r in the equation is for position and by relevant equations you mean the equations of motion? and even if it is so how do I use them?

4. May 27, 2017

### haruspex

Yes, but that makes statement a) vacuous. I think you can reasonably assume it means in some given common frame.
Yes, and you could just answer d), as you have, but if you want proof that the other answers don't work then equations are going to help. You don't necessarily have to solve any, just make deductions from them.
Take c). Introduce variables for their respective velocities wrt that distant star and for the velocities of the particle they observe. What equation can you write?

5. May 28, 2017

### Rahulrj

Just getting the answer would not get me to learn anything so I would want to know why others are wrong. If I name the velocities as $v_1$ for the observer who sees the particle at constant velocity and $v_2$ for the observer who sees the accelerating particle how can I write write the equations for the case $c$?

6. May 28, 2017

### haruspex

As I wrote, you need two more velocity variables - the two velocities that the two observers ascribe to the particle.

7. May 28, 2017

### Rahulrj

oh okay I missed that part. So the velocity of particle wrt observer 1 is $v_{1p}$ and that wrt observer 2 is $v_{2p}$. So now how do I form the equations relevant to each?

8. May 28, 2017

### haruspex

You have two ways of expressing the velocity of the particle relative to the star.

9. May 28, 2017

### Rahulrj

I often find myself confused when describing relative motion particularly when there is no directions given because I am not sure what velocities are to be subtracted/added from the given,would be really helpful if you could give some essential points to keep in mind. Anyway this is what I have come up with $v_{rel1} = v_1 - v_{1p}$ $v_{rel2} = v_2 - v_{2p}$ where $v_{rel1}$ and $v_{rel2}$ represents relative velocity of particle wrt star.

10. May 28, 2017

### Aryamaan Thakur

a) both observers are at rest
b) second observer is moving with constant velocity w.r.t the first observer
c) both observers are moving with constant velocity w.r.t to a distant star
d) second observer is accelerating w.r.t first observer

1. if both are at rest the object will seem to move with same velocity for both of them
2. okay, this one satisfies the first condition but a/q the object is accelerating so if the second observer is moving with a constant velocity wrt first he/she will see the object with a constant velocity and not accelerating
3. again both will see the object moving with same velocity
4. Yes, this seems to be right. If the object is moving with constant velocity wrt to first observer and the second observer is accelerating wrt first observer then ultimately the object is accelerating wrt second observer.

So, the right answer is (d)

I hope it will help

11. May 28, 2017

### haruspex

You've not written them as vectors, but I'll assume that is implied.
You are not combinng the relative velocities correctly. B relative to A plus C relative to B gives C relative to A.
There is only one velocity of the particle relative to the star, which is how you get an equation relating the four other relative velocities.

12. May 28, 2017

### Rahulrj

Not sure if I am confusing things now but Wouldn't there be two relative velocities of particle relative to star wrt to each observer? for example, V of particle relative to star = V of observer 2 wrt to star + V of particle wrt obs: 2. ($v_{rel2} = v_2 + v_{2p}$ )

13. May 28, 2017

### haruspex

No, the velocity of the particle relative to the star is independent of (reliable) observer in Newtonian mechanics.

14. May 28, 2017

### Rahulrj

So then is this relation incorrect? (V of particle relative to star = V of observer 2 wrt to star + V of particle wrt obs: 2).how then can I relate both the observers and relative velocity of particle wrt star in one equation?

15. May 28, 2017

### haruspex

Yes.
Just write the same equation going via observer 1 instead of observer 2.
You do not actually care about the velocity of the particle relative to the star, but it is what connects the variables you do care about.

16. May 28, 2017

### Rahulrj

That exactly was my doubt here that if I can write two equations of relative velocity wrt each observer as opposed to what you said that there is only one velocity of particle relative to star. So then the equation will be $v_{rel1} = v_1+v_{1p}$

17. May 28, 2017

### haruspex

You can write one expression for the velocity of the particle relative to the star using the first observer, and a second expression for that same velocity using the second observer (as you did in post #14). Because those two relative velocities (particle relative to star) are necessarily the same, the two expressions are equal. That is the equation you need.

18. May 28, 2017

### Rahulrj

Yes I figured that in your post #15. I was stuck when you said there is only one velocity of particle relative to star because I misunderstood taking it be that there is only one expression. So now it will be, $v_2+v_{2p} = v_1+v_{1p}$ and for the case in $c$, $v_1=v_2$ and therefore $v_{2p}=v_{1p}$. So similarly I have to form an expression for the case $b$ where observer 2 moves at a velocity wrt observer 1?

19. May 28, 2017

### haruspex

Yes.
The same equation applies in all cases. It makes no assumption about whether the velocities are constant or variable.
What do you need to do to the equation to get one relating to accelerations?

20. May 28, 2017

### Rahulrj

That's simple, I just have to take the derivative of v for acceleration. In the case of b, I am not sure how the same equation can apply they are taken wrt to star but the case is wrt to observer 1 making no mention of a star. So shouldn't the expression be (V of particle wrt obs 1 = V of particle wrt obs 2+ V of obs 2 wrt obs 1)? but I am not sure what is the common relative velocity here.