| Topic: |
Science > Physics |
| User: |
"Chris Wood" |
| Date: |
30 Jan 2004 03:47:57 AM |
| Object: |
Stumped by Basic Special Theory Concept Question |
I teach Physics at a community college.
We're covering Special Relativity. Our
textbook included a concept question
that has me, and my students, stumped.
It's driving us crazy. I think it's a classic
Special Relativity question.
----
A rocket moves toward a mirror at 0.800c relative to
a fixed reference frame S. The mirror is stationary
relative to S. A light pulse emitted by the rocket
travels toward the mirror and is reflected back to
the rocket. The front of the rocket is 1.40x10^12 m
from the mirror (as measured by observers in S)
at the moment the light pulse leaves the rocket.
What is the total travel time of the pulse as
measured by observers in the S frame?
What is the total travel time of the pulse as
measured by observers in the front of the rocket?
.
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| User: "George Jones" |
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| Title: Re: Stumped by Basic Special Theory Concept Question |
31 Jan 2004 01:32:00 PM |
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Chris Wood wrote:
I teach Physics at a community college.
We're covering Special Relativity. Our
textbook included a concept question
that has me, and my students, stumped.
It's driving us crazy. I think it's a classic
Special Relativity question.
----
A rocket moves toward a mirror at 0.800c relative to
a fixed reference frame S. The mirror is stationary
relative to S. A light pulse emitted by the rocket
travels toward the mirror and is reflected back to
the rocket. The front of the rocket is 1.40x10^12 m
from the mirror (as measured by observers in S)
at the moment the light pulse leaves the rocket.
What is the total travel time of the pulse as
measured by observers in the S frame?
What is the total travel time of the pulse as
measured by observers in the front of the rocket?
The first question can be answered without using time dilation, Lorentz
contraction, Lorentz transformations, invariance of the interval etc.
The only concepts necessary are:
1) the speed of light in S is c, even though the light is emitted by a
rocket moving with respect to S;
2) delta x = v delta t.
Call the emission of the light by the rocket event A, the reflection of
the light by the mirror event B, and the reception of the light by the
rocket event D. I always like to draw a spacetime diagram first. I have
chosen event A as the origin of the spacetime diagram, and note that
xB = 1.40x10^12 m is known.
ct
| D / |
| /. |
| / . |
| / . |
| / .|B
| / . |
| / . |
| / . |
| / . |
| / . |
| / . |
| / . |
| / . |
|/.__________________|_____________ x
A
The rocket is represented by the sloped dashed line, light by the dotted
lines, and the mirror by the vertical dashed line at the right.
light traveling from A to B:
(delta x)/(delta t) = c gives xB - xA = (tB - tA)c with xA = tA = 0
tB = xB/c (1)
light traveling from B to D:
delta x)/(delta t) = -c gives xD - xB = -(tD - tB)c (2)
rocket traveling from A to D:
(delta x)/(delta t) = v gives xD - xA = (tD - tA)v with xA = tA = 0
xD = tD*v (3)
Using (1) and (3) in (2) gives
tD*v - xB = -(tD - xB/c)c (4)
Solve (4) for tD.
tD*(c+v) = xB*2
tD = xB*2/(c + v)
= xB*(10/9)/c (5)
To answer the second question, I would use invariance of the interval,
as it is more fundamental then either time dilation or Lorentz
transformations.
Above, I have used unprimed coordinates for the S frame. Continue doing
this, and now also use primed coordinates for the rocket frame. The
answer to your second question is tD'. Invariance of the interval means
(delta x)^2 - (c*delta t)^2 = (delta x')^2 - (c*delta t')^2
for any 2 events. Choose events A and D, and note that since the rocket
doesn't move in its own frame, delta x' = 0.
Invariance of the interval with 0 = xA = tA = xA' = tA' = delta x':
xD^2 - (c*tD)^2 = - (c*tD')^2 (6)
Use (3) and (5) in (6):
(v*tD)^2 - (c*tD)^2 = - (c*tD')^2
tD' = tD*sqrt(1 - v^2/c^2) (7)
= tD*(3/5)
= xB*(2/3)/c
If you (and/or the text) don't cover invariance of the interval, then
(7) can also be arrived at via time dilation. I usually teach time
dilation as a concept and as an application of invariance of the
interval, but I don't teach it as a method for solving problems.
Standard disclaimer: I don't guarantee that any of the above is
correct.
Regards,
George
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| User: "Sam Wormley" |
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| Title: Re: Stumped by Basic Special Theory Concept Question |
30 Jan 2004 07:55:27 AM |
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Chris Wood wrote:
I teach Physics at a community college.
We're covering Special Relativity. Our
textbook included a concept question
that has me, and my students, stumped.
It's driving us crazy. I think it's a classic
Special Relativity question.
----
A rocket moves toward a mirror at 0.800c relative to
a fixed reference frame S. The mirror is stationary
relative to S. A light pulse emitted by the rocket
travels toward the mirror and is reflected back to
the rocket. The front of the rocket is 1.40x10^12 m
from the mirror (as measured by observers in S)
at the moment the light pulse leaves the rocket.
What is the total travel time of the pulse as
measured by observers in the S frame?
What is the total travel time of the pulse as
measured by observers in the front of the rocket?
What calculations have you done to date?
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| User: "EjP" |
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| Title: Re: Stumped by Basic Special Theory Concept Question |
30 Jan 2004 08:48:35 AM |
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Chris Wood wrote:
I teach Physics at a community college.
We're covering Special Relativity. Our
textbook included a concept question
that has me, and my students, stumped.
It's driving us crazy. I think it's a classic
Special Relativity question.
----
A rocket moves toward a mirror at 0.800c relative to
a fixed reference frame S. The mirror is stationary
relative to S. A light pulse emitted by the rocket
travels toward the mirror and is reflected back to
the rocket. The front of the rocket is 1.40x10^12 m
from the mirror (as measured by observers in S)
at the moment the light pulse leaves the rocket.
What is the total travel time of the pulse as
measured by observers in the S frame?
What is the total travel time of the pulse as
measured by observers in the front of the rocket?
Solving the problem in S:
Key fact: light always travels at c in a particular
frame. Use this to calculate the time it takes
to travel each leg.
The first leg is trivial.
The rocket fires the mirror from a distance L1=L away,
so it takes t1 = L/c to reach the mirror. You
can calculate this directly.
To calculate the return trip, you have to realize
that the rocket has moving toward the mirror
at a constant velocity the whole time, so the total
distance the light travels on the return trip is
the *original* distance minus the distance the
ship has moved. In other words, L2 = L - .8c*T,
where T is the *total* time. You can express T
as the sum of the two legs and get:
t2 = T-t1 = L2/c
-> T-t1 = (L - .8*c*T)/c = L/c - .8*T
The only unknown is T, and you can solve for it.
At this point, you can solve to the time in the
ship in one of two ways:
- Realize that it represents the "proper time" of
it's frame, so the elapsed time will be T' = T/gamma
- Solve the whole problem again starting with
the distance as measured in the rocket frame:
L' = L/gamma.
You'll get the same answer, of course.
-E
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| User: "Gregory L. Hansen" |
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| Title: Re: Stumped by Basic Special Theory Concept Question |
30 Jan 2004 09:12:33 AM |
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In article <3Y6dnbAXXtr-tYfd4p2dnA@comcast.com>,
Chris Wood <cjwood99@comcast.net> wrote:
I teach Physics at a community college.
We're covering Special Relativity. Our
textbook included a concept question
that has me, and my students, stumped.
It's driving us crazy. I think it's a classic
Special Relativity question.
----
A rocket moves toward a mirror at 0.800c relative to
a fixed reference frame S. The mirror is stationary
relative to S. A light pulse emitted by the rocket
travels toward the mirror and is reflected back to
the rocket. The front of the rocket is 1.40x10^12 m
from the mirror (as measured by observers in S)
at the moment the light pulse leaves the rocket.
What is the total travel time of the pulse as
measured by observers in the S frame?
What is the total travel time of the pulse as
measured by observers in the front of the rocket?
Find the equations of motion for the light in the frame S, and identify
the times and locations that the light hits the mirror and returns to the
rocket. The light pulse is emitted at (t1,x1), hits the mirror at
(t2,x2), returns to the rocket at (t3,x3). This is a freshman "when does
the leapord catch the gazelle" type of problem.
Then use the Lorentz transforms to find what those events are in the
rocket frame.
Actually, since the light pulse is emitted and received by the same
rocket, x1'=0 and x3'=0, I'm getting a simple application of time
dilation.
Or boost directly to the rocket frame. In S, the mirror's equation of
motion is x(t)=L. In S', the mirror's equation of motion is
x'(t')=(l-vt')/sqrt(1-v^2/c^2). The light still goes at c. Then it
becomes just another leopard problem, and you should get the same answer.
--
"Are those morons getting dumber or just louder?" -- Mayor Quimby
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| User: "Alex Leibovici" |
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| Title: Re: Stumped by Basic Special Theory Concept Question |
30 Jan 2004 12:50:23 PM |
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You can avoid splitting the movement into two different stages (to and
from the mirror) by simply considering the movement of the light pulse
toward the rocket's _image_.
In that case you have two bodies, initially at a distance L=twice the
given number, which travel one towards another with the speeds "c" and
0.8*c. There is no relativity here, just arithmetics !
Time will be then = L/(c+0.8*c)
There is no relativity here, just arithmetics !
Note that if two light pulses travel one toword another, they will
meet after L/(c+c) seconds.
Alex
"Chris Wood" <cjwood99@comcast.net> wrote:
I teach Physics at a community college.
We're covering Special Relativity. Our
textbook included a concept question
that has me, and my students, stumped.
It's driving us crazy. I think it's a classic
Special Relativity question.
----
A rocket moves toward a mirror at 0.800c relative to
a fixed reference frame S. The mirror is stationary
relative to S. A light pulse emitted by the rocket
travels toward the mirror and is reflected back to
the rocket. The front of the rocket is 1.40x10^12 m
from the mirror (as measured by observers in S)
at the moment the light pulse leaves the rocket.
What is the total travel time of the pulse as
measured by observers in the S frame?
What is the total travel time of the pulse as
measured by observers in the front of the rocket?
Alex Leibovici
-------------
"You cannot reason a person out of something he has not been
reasoned into." Jonathan Swift
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| User: "Edward Green" |
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| Title: Re: Stumped by Basic Special Theory Concept Question |
31 Jan 2004 07:39:22 PM |
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Alex Leibovici <ALeibovici@solnet.ch> wrote in message news:<b79l10tbkfbbsogra3ebf8huqsmhsdan7f@4ax.com>...
You can avoid splitting the movement into two different stages (to and
from the mirror) by simply considering the movement of the light pulse
toward the rocket's _image_.
In that case you have two bodies, initially at a distance L=twice the
given number, which travel one towards another with the speeds "c" and
0.8*c. There is no relativity here, just arithmetics !
Time will be then = L/(c+0.8*c)
There is no relativity here, just arithmetics !
I prefer to say, there is no dynamics here, just kinematics.
It really gets some people excited that ordinary kinematics continues
to work -- quite blase about special relativity -- in any fixed frame
of reference.
Note that if two light pulses travel one toword another, they will
meet after L/(c+c) seconds.
No! That's obviously wrong! You have not used correct relativistic
addition of velocities!!! ;-)
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| User: "Steven Gray" |
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| Title: Re: Stumped by Basic Special Theory Concept Question |
01 Feb 2004 08:59:20 AM |
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(Edward Green) wrote in
news:2a0cceff.0401311739.63afb71@posting.google.com:
Note that if two light pulses travel one toword another, they will
meet after L/(c+c) seconds.
No! That's obviously wrong! You have not used correct relativistic
addition of velocities!!! ;-)
Careful! You need to be sure that you've defined exactly what is meant
here. In my own inertial frame of reference, suppose one stationary
light source is located 1 mile away from me in one direction and another
is located 1 mile away in the opposite direction. The distance between
them is 2 miles. If they both send out a light pulse at the same time,
the pulses do meet in L/(2c) seconds, where L is 2 miles. The "closing
speed" of the two pulses, as measured by me in my own frame of reference,
is 2*c.
--
Steve Gray
sgray2@cfl.rr.com
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| User: "Alex Leibovici" |
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| Title: Re: Stumped by Basic Special Theory Concept Question |
30 Jan 2004 01:00:05 PM |
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Alex Leibovici <ALeibovici@solnet.ch> wrote:
You can avoid splitting the movement into two different stages (to and
from the mirror) by simply considering the movement of the light pulse
toward the rocket's _image_.
To be on the safe side, by "image" I mean here the geometrical, not
the real image of the rocket. Have still to figure out what the real
(?) image will do . . . . ;-)
Alex Leibovici
-------------
"You cannot reason a person out of something he has not been
reasoned into." Jonathan Swift
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