Reference frames and Galilean transformation

[Taylor_1989]

Homework Statement

I am having a issue relating part of this question to the Galilean transformation.

Question

Relative to the laboratory, a rod of rest length $l_0$ moves in its own line with velocity u. A particle moves in the same line with equal and opposite velocity . How long dose it take for the particle to pass the rod.

Homework Equations

$v=v'+w$

The Attempt at a Solution

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So my issue is relating the velocities to the gaillen transformation. My ans from a) , b) and c) for velocity seen in each ref frame is $2u$, which has been workout from the right side of the diagram, but my issue is relating to the left side diagrams which are Galilean where $S$ the stationary frame and $S'$ is the moving ref frame.

So more first diagram if I use the gallien equation displayed in the relevant equations then I make the velocity $-2u$ comapred to the middle where I make it $2u$ and for the last I make it $0$ which is obviously not correct. This has always been my issue, when relating to the gaillen transformations, I just can never figure what equations to use for a give situation, but if I do it like have on the left side I have a better understanding.

I was just wondering if someone could advise me on where I seem to go wrong.

 

[PeroK]

You are given the velocities in the lab frame, so your last diagram showing the lab frame is correct.

Can you solve the problem using just the lab frame? That will give you the answer.

If you transform to another frame, you must make the same transformation to all objects. In your first two diagrams you appear to made different transformations to different objects.

 

[Taylor_1989]

Can you solve the problem using just the lab frame? That will give you the answer.

I am not sure what you mean by this, as to me if I am in the frame of the rod, the lab would be appear to moving at a velocity of $u$ in the opposite direction to travel of the rod.

 

[PeroK]

I am not sure what you mean by this, as to me if I am in the frame of the rod, the lab would be appear to moving at a velocity of $u$ in the opposite direction to travel of the rod.

Why do you need to think about the rod frame to solve the problem? Can't you just use the lab frame?

 

[Taylor_1989]

Ok so I have made some correction to my diagram, and using ur comment:

Can you solve the problem using just the lab frame?

I now have the following solutions, from my diagram.

$a) 2u ms^{-1}$

$b) 2u ms^{-1}$

$c) 2u ms^{-1}$

 

[PeroK]

It's difficult to see what you've done and what you are trying to calculate.

 

[Taylor_1989]

It's difficult to see what you've done and what you are trying to calculate.

So for my top diagram, I have used the gaillen transformation of $v'_{particle}=v_{particle}-v'_{rod frame}-$.

So if I take anything moving to the right as positive then $S'$ frame will move at the velocity of what the rod was moving at in $S$ and then the particle velocity in the $S'$ will be moving to the left $-v'_{particle}$ so then I found the ans for a) to be $-v_{particle}=-2ums^{-1}\rightarrow v'{particle}=2ums^{-1}$ .

so the particle will pass the rod in $t=l/2u$

Is this a little clearer?

 

[PeroK]

So for my top diagram, I have used the gaillen transformation of $v'_{particle}=v_{particle}-v'_{rod frame}-$.

So if I take anything moving to the right as positive then $S'$ frame will move at the velocity of what the rod was moving at in $S$ and then the particle velocity in the $S'$ will be moving to the left $-v'_{particle}$ so then I found the ans for a) to be $-v_{particle}=-2ums^{-1}\rightarrow v'{particle}=2ums^{-1}$ .

so the particle will pass the rod in $t=l/2u$

Is this a little clearer?

Yes. So what's the problem?

 

[Taylor_1989]

What I am asking have I understood how to form the gallien transformation correctly?

 

[PeroK]

What I am asking have I understood how to form the gallien transformation correctly?

I get the impression you're not sure conceptually what you're trying to do. In this problem you are dealing with a 1D velocity transformation.

I'm also not sure whether you could have got that answer in your Head and were struggling with the formal transformation of coordinates or whether you don't intuitively see how velocities add.

 

[Taylor_1989]

I get the impression you're not sure conceptually what you're trying to do. In this problem you are dealing with a 1D velocity transformation.

I'm also not sure whether you could have got that answer in your Head and were struggling with the formal transformation of coordinates or whether you don't intuitively see how velocities add.

Ah, Okay. My issue was I could not see how the formal transformation work I can see how the ans work in my head but it trying to show them as formual transformations as in draw what I see in my head as Galilean transformations. I ma not have explained this fully in OP. I have always sturggled trying to related what I see in my head on the transformation graphs.