Calculation 1
Example 1
For Example 2 See: Calculation 2
This example is used to explain the mathematics involved in length contraction
Consider a spaceship with a length l0 = 300000 km
This spaceship has a speed v = 0.999*c
There is one observer at the back of the spaceship and one at the front of the space ship.
At the back of the spaceship a light pulse is transmitted to the front and reflected to the back.
Questions:
 In the rest frame, when will the pulse reach the front of the spaceship?
 In the rest frame, when will the pulse reach the back of the spaceship?
 In the moving frame, when will the pulse reach the front of the spaceship?
 In the moving frame, when will the pulse reach the back of the spaceship?
 With v=0, when will the pulse reach the front of the spaceship?
 With v=0, when will the pulse reach the back of the spaceship?
 What if no length contraction is involved?
In questions 1 and 2 clocks in the rest frame are used. In questions 3 and 4 clocks in the moving spaceship are used.

  / /
t1+t2.  t1+t2. /
 .  / . /
 .  / . /
 .  / . /
 .  / .t1
 .  / ./
 .  / . /
 .  / . /
 .t1 / . /
 .  / . /
 .  / . /
 .  / . /
 .  / . .t4
 .  / . ./
 .  / . . /
 .  /. . / >
t3..t4t3../t0
 . .  / . . /
 . .  B.F
 . . 
B.F
v=0 v>0
B = Back of the train or spaceship
F = Front of the train or spaceship
. = Light signal
 
The above left figure shows the situation for v = 0 (The questions 5 and 6)
The above right figure shows two aspects of this example for v>0 (The questions 14)
Each figure is subdivided into two parts.
 The top part above the line t0 shows the example:
 At t0 (=t3) there is a light signal at the back of the train.
 This light signal reaches the front F at the train at t1.
 The light signal reaches the back B at the train at t1+t2
 Delta time between the back and the front signals is t2
 The bottom part shows clock synchronization.
 Clock synchronization starts with two light signals at the middle from the train.
 One signal reaches the back B of the train at t3. This signal is the start signal t0 of the example.
 The other signal reaches the front B of the train at t4.
 The two signals t3 and t4 are used by the observers on the train to reset their watches.
Answers

First of all length contraction is involved.
That means the length of the moving train is not l0 but l (measured in the rest frame).
l = l0 * sqrt(1v*v/c*c) = l0 / gamma
gamma = 1 / sqrt(1v*v/c*c)
 The last sixth question is the easiest:
Total duration in rest frame with speed is zero = 2 * l0 / c = 2 Secs.
 The answer for the fifth question is:
The duration going from back to front is 1 Second.
 In order to answer the first question you must use the formula:
l + v*t = c*t or
l = c*t  v*t or
t1 = t = l / (cv)
 In order to answer the second question you must use the formula:
l = v*t + c*t or
t2 = t = l / (c+v)
The total time since t0 = 0 is equal to t1+t2
t1+t2 = l / (cv) + l / (c+v)
 In order to answer the fourth question you must divide the previous time t1+t2 by gamma. This takes Time Dilation into account i.e. that the moving clock runs slower.
 In order to answer the third question you must first synchronise the two clocks on board of the train.
The time t3 that one signal reaches the front = 0.5*l/(c+v)
The time t4 that one signal reaches the back = 0.5*l/(cv)
If the watch of the observer at F is reseted at t4 then the time from the back of the train does not reach the observer at t1 but at t1(t4t3) in rest frame
In the reference frame of the moving train you have to divide that value by gamma
In order to do the example for a particular value of v/c you can use the following calculator:
Reflection
The last value of line three and the last value of line four demonstrates, that the observations of the observers in the moving train are identical for those observers as when the train stands still.
The observations are independent of the speed v of the train, in accordance with Special Relativity.
Line five, the values when no Length Contraction is involved, is not in accordance with Special Relativity.
Physical Interpretation
If the results of actual experiments are in accordance with the lines 1 to 4 than this implies that Length Contraction is a physical effect.
The same for Time Dilation.
If the results of experiments and or observations are in accordance with the line 5 than no Length Contraction is involved.
Feedback
None
Created: 8 February 2002
Modified: 19 November 2004
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