Question

Standing beside railroad tracks, we are suddenly startled by a relativistic boxcar traveling past us as shown in the figure. Inside, a well-equipped hobo fires a laser pulse from the front of the boxcar to its rear. (a) Is our measurement of the speed of the pulse greater than, less than, or the same as that measured by the hobo?

Answer 1

The pulse travels at the speed of light, which is the same for either observer.

Answer 2

Tis a postulate of relativity! The speed of light c is the same in every reference frame

Answer 3

haha thanks ! next it asks about the time taken by the pulse to reach from front to end

would be be same for both the observers ?

Answer 4

this is supposed to be a qualitative question
im hoping to answer this using the postulate of relativity w/o doing too much math..

Answer 5

Wouldn't the time be shorter for the outside observer, because of the length contraction of the train car...

Answer 6

oh my textbook hasn't talked length contraction yet...
so far it covered the relativity postulate and the relativity of simultaneity

Answer 7

According to Einstein's Special Theory of Relativity, the speed of light is the same when measured in any inertial reference frame. :)

And so as for the time it takes for the pulse to travel that far, note that there will be relativistic effects when it comes to the time.

Answer 8

Are you familiar with time dilation?

Answer 9

somewhat...

Answer 10

i cannot use this equation directly here as the events are not local in either frames of referencce
\[\Delta t = \dfrac{\Delta t_0}{\sqrt{1-(v/c)^2}}\]

right ?

Answer 11

By "not local", i mean, the events are "spatially separated" :

we're taking timestamp at different positions in space

Answer 12

\(\color{blue}{\text{Originally Posted by}}\) @ganeshie8
i cannot use this equation directly here as the events are not local in either frames of referencce
\[\Delta t = \dfrac{\Delta t_0}{\sqrt{1-(v/c)^2}}\]

right ?
\(\color{blue}{\text{End of Quote}}\)
Splendiforous!

Answer 13

But as far as I know, this event is indeed local. The light's traveling from the hobo to the other side of the train is something which occurs on the train.

Answer 14

\(\color{blue}{\text{Originally Posted by}}\) @ganeshie8
By "not local", i mean, the events are "spatially separated" :

we're taking timestamp at different positions in space
\(\color{blue}{\text{End of Quote}}\)
Hmmm.

Answer 15

yeah but dont we need to use two different clocks positined at two different places :
1) front of the trian
2) end of the train

Answer 16

that changes the time dilation equation that i have posted earlier right ?

Answer 17

while deriving that time dilation equation, the textbook used "only one clock" in the train frame of reference. so im wondering how to change that equation to answer the present question...

Answer 18

`Wouldn't the time be shorter for the outside observer, because of the length contraction of the train car...`

that time dilation equation says the exact opposite right ? @agent0smith

Answer 19

\(\color{blue}{\text{Originally Posted by}}\) @ganeshie8
that changes the time dilation equation that i have posted earlier right ?
\(\color{blue}{\text{End of Quote}}\)

it doesn't make a difference, because in a given frame, all the clocks are synchronized

Answer 20

\(\dfrac{1}{\sqrt{1-(v/c)^2}}\gt 1\) for all \(v\gt 0\)

Answer 21

so the two clocks in the train frame are working identically, and they are synchornized to some "master clock" inside the train

Answer 22

If you want, you can find the delta t for each frame separately too and you'll see that they're related by the time dilation equation

Answer 23

do you want me to demonstrate that?

Answer 24

yeah, please...

Answer 25

Lets start with the train frame: for an observer in the train, the distance covered is L (length of the train compartment) and speed is c so time is:

\[\Delta \space t \space train \space = \frac{ L }{ C }\]

Answer 26

Okay

Answer 27

the person on the ground sees the length of the train contracted

Answer 28

why/how ? this is my first time reading relativity, please be slow...

Answer 29

ok so the person on the ground sees the length of the train compartment to be:
\[\sqrt{1- \frac{ v }{ c^2 }}\]

thats the Lorentz contraction formula

Answer 30

\(\color{blue}{\text{Originally Posted by}}\) @ganeshie8
why/how ? this is my first time reading relativity, please be slow...
\(\color{blue}{\text{End of Quote}}\)
oh you mean why is it L/C in the train frame?

Answer 31

no, im talking about the mess right after that..

Answer 32

length contraction and lorentz formula..

Answer 33

how are they consequences of the relativity postulate ?

Answer 34

lorentz contraction is a consequence of the postulates of relativity, just like time dilation is

Answer 35

the proof is long, but you can find it in any textbook

Answer 36

il need to study length contraction to answer present question is it ?

Answer 37

its not essential for this question, because you can just answer it from the time dilation formula

Answer 38

I was just showing you an alternative methd of getting the same answer

Answer 39

but its not necessary to use Lorentz contractions for this problem

Answer 40

okay, using time dilation equation im getting this :

\[\text{interval measured by outside observer} \\= \dfrac{\text{interval measured by inside observer} }{\sqrt{1-(v/c)^2}}\]

Answer 41

\(v\) = relative velocity between outside and inside observers
\(c\) = speed of light

Answer 42

Correcto

Answer 43

you have "outside observer" for both sides...which one did you mean for the train?

Answer 44

since \(\dfrac{1}{\sqrt{1-(v/c)^2}}\) is always greater than \(1\),
the outside observer measures more time compared to the inside observer ?

Answer 45

Correcto

Answer 46

ive corrected that...

Answer 47

basically "time slows down" for the inside observer

Answer 48

for the same event, the inside observer is measuring shorter time compared to the outside observer

Answer 49

we have to be careful about terminology in relativity - what do you mean by "event"? :O

Answer 50

im talking loosely
but the textbook definition is event is start / end of a process with a specific timestamp

Answer 51

here, both observers measured the time elapsed between same two events.

and we concluded that the inside observer's measured time is less than the outside observer's measured time

Answer 52

Yep, pretty much.

Answer 53

Iʻm the hobo, on the train

Answer 54

Hi hobo

Answer 55

I think when you read about simultaneity, it was not justified, keep in mind synchronizing clocks at different places are relative and not absolute. A good example maybe, here we have 3 observation stations a,b, and c (equally spaced) on the x - axis of the inertial frame S where they are all at rest, the red lines is often used in special relativity which are called world lines, which is just a function of its position as a function of time. Note the lines are vertical, and now say a flash light signal is sent out from b at @ t = 0 it travels back and forward as speed along the x - axis. Where the intersection will happen at a1 and c1 so we can see a and c are defined by the line a1c1.


So far so good? It gets tricky after this point, I don't know if I should wait till you've read a bit more or keep going :d