The resolution of these small original exercises will give you a better mastery of Lorentz= formulae, will allows you to better understand their meaning, will allows you to see that they are perfectly interpreted in the framework of classical mechanics

It is advised to have made previously exercises on time zones.
Exercises using any speed (greater than c) can be solved simply.

Let S an Ox axis, S is a classical axis.
In each point of this axis is found a number, its abscissa, and is found a clock that indicates the date of any event happening on this point.
Clocks of S are classical clocks, they are all synchronous.

Let S' another O=x= axis sliding on Ox at constant v speed , relatively to S.
The equation of the motion of O' , relatively to S , is x = vt.
In each point of S' is found a number x', its abscissa, and is found a clock that indicates dates, relatively to S', of events that happens on x'.
Each point of S' has its clock.
And more one supposes that one passes from S to S' with Lorentz= transformations of constant c, c > abs (v), abs meaning  absolute value.

Let us take for example an explosion that has for coordinates (x, t) , relatively to S .
The coordinates (x', y=) of this same explosion, relatively to S' , are given by Lorentz= transformations.

Thus by : x' = (x - vt) / b ; y' = (t - vx / c^2) / b , with b = sqrt (1-v^2 /c^2)  (sqrt meaning squared root)

Lets S" a third  O"x" axis that slides on Ox at constant w speed, relatively to S, with abs(w) < c, c being the above positive constant.
The equation of the motion of O" , relatively to S , is x = wt.
One supposes that one passes again from S to S" by Lorentz= transformations with replacing v by w in preceding formulae.

Exercise 1.

One takes c = 5, v = 3, and w = 4.

1. Represent axis S, S', S" at the dates t = 0 , then t =1, then t =2, then t = 3 of S.
    One will take 25 mm for one unit on Ox.
    One will place relative integer abscissas on the three axis, and one will place corresponding clocks.

2. Using drawings only, find the speed of O' , relatively to S" , then find the speed of O" , relatively to S'.

3. Let A' the point of S' of abscissa x' = 4 + 1/16.
    Represent A' on the drawings corresponding to t = 0, then to t = 1.
    Place abscissas and clocks of S" found on A' .
    Then deduce graphically, the measure of O'A= , relatively to S".
    (Take care to use correctly the definition of simultaneity, relatively to S" )

 
Exercise 2

In this exercise, 0 < v < c

Let M a moving point on S .
The equation of M, relatively to S , is x = kt , k any real constant.
(The speed of M is k, relatively to S, and abs(k) can take values greater than c ).

1. Determine the equation of the motion of M , relatively to S' , then give the speed k' of M , relatively to S'.
    For what value of k, k' does not exist?  How then is perceived M in S' ?

2. One supposes k > c.
    Discuss, in function of k, the value of  k' , and the sign of k'.
    Show especially that from a given value of k, k and k' are of opposite signs.
    What is limit of k', when k tends tower plus infinite ?
    Is S' a system of reference always coherent ?

3. Verify that any negative value of k , gives a negative value of k' .
    Show that if k<-c then k'<-c .
    What is limit of k', when k tends toward minus infinite?
    Compare with the result of the question 2.

4. Let a ruler of length L , relatively to S , sliding on S at k speed, k relative to S.
    One supposes that the respective equations of origin and extremity of this ruler are:

                       x = kt      and      x = kt + L.

                        k being any real constant.

     a) Determine the length L' of this ruler , relatively to S'.

     b) Discuss in function of k and L, the sign and the value of L'.
         One will specify especially the value of k for which L= is infinite.

     c) What is limit of L', when k tends toward minus infinite ?
         When k tends toward plus infinite ?

5. Give interval of values of k , for which S' remains a coherent system.

6. This last question is more difficult because not detailed.
    A moving point N slide on S according to equation x = t.t (t.t means squared t).
    The motion of N is brought to S.
    Give its equation in S'. Discuss about this equation .
    Explain the pseudo-paradoxes obtained.
    Specify the speed of N , relatively to S' . Discuss result.

Exercise 3

The aim of this exercise is to see that the constant c in Lorentz' transformations is not an absolute.
Can coexist different systems obeying to Lorentz' formulae , c taking different values according to the systems.
Can coexist at  same time, Galilean systems.

In this exercise one passes from S to S' by Lorentz' transformations of constant c , and one passes from S to S" by Lorentz' transformations of constant c" .
c and c" are two strictly positive reals.
One passes from S to Sg', and from S to Sg" , by Galilean' transformations.

The speed of S' , relatively to S , is v . (abs(v)<c).
The speed of S", relatively to S , is w . (abs(w)<c").
The speed of Sg', relatively to S , is v .
The speed of Sg", relatively to S , is w .
At  t = 0 , axis S , Sg' , Sg" are coinciding .

1. Let an explosion of coordinates (x, t) , relatively to S,
    of coordinates (x', y'), relatively S', of coordinates (x", t") , relatively to S" ,
    of coordinates (xg',yg'), relatively to Sg' , and of coordinates (xg",yg") , relatively to Sg".

     a) Write (x', y') in function of (x, t), then write (x", t") in function of (x, t).

     b) Write (x", t") in function of (x', y'), then write (x',y') in function of (x", t").

     c) Write (x',t') in function of (xg',yg') then in function of (xg",yg") .

2.  a) Determine the speed of S' relatively to S" .
         Determine the speed of S" relatively to S' .
        Are these two speeds opposite ?
        One can discuss in function of v, w, c, c" .

     b) Determine the speed of S" , relatively to Sg', then the speed of S' , relatively to Sg".
        Are these two speed opposite ?

3. Let a moving point of equation x = kt , relatively to S .
   Let k' and k" respective speeds of this point in S' and in S" .

   Express k" in function of k', then k' in function of k".

4. Let R a ruler of (algebraic) length L , relatively to S , and moving at k speed, relatively to S .
    Let L' the length of R , relatively to S' .
    Let L" the length of R , relatively to S" .

    a) Express L" in function of L' , then L' in function of L".

    b) For what value(s) of k , can we have L" = L' ? Is it always possible ?

5. Let two explosions E1, E2 , simultaneous relatively to S' , and distant of 5 units, relatively to S'.

    Is it possible to have, relatively to S" , these two explosions simultaneous ?
    If yes, give non evident solutions. (w=v and c=c" is an evident solution).

Exercise 4.

To solve this exercise, one can be helped by results given in "Elementary Study of Lorentz= Transformations .
We always stay in the framework of the classical mechanics.

Let Ox an infinite elastic band.
One supposes this elastic being perfect.

Every transversal wave on this elastic band has an equation of the form

                     y=f (t- x/c) + g (t+x/c)

f and g being two any functions of one variable.
c is a strictly positive constant, determined by elastic characteristics of the band.
c is transversal speed of the free waves on this band.

Let v a real verifying abs (v) < c (abs absolute value), and let x' and t' the transformed of x and t by Lorentz= transformations, transformations in which v and c are constants of this exercises.

1. Demonstrate that y=f(t'- x'/c) + g (t'+ x'/c) is a possible transversal wave on the same elastic band.

2. Let f a periodical function of period p > 0.
   Determine all motionless points of the (O) wave defined by:

                    y = f (t - x/c) - f (t+x/c)

   (It is evident that x=0 is a permanent point).

   What is the distance (noted d) between two consecutive permanent points?

3. Let the wave (noted (Ov)) defined by :

                 y = f(t'-x'/c) - f(t'+x'/c)

   Determine the smallest positive constant (noted L), such that for all t, the point
   of abscissa  x = vt + kL  (k relative integer) has zero for ordinate .

4. The wave (O) has a vibratory period of p (the elastic band finds again the same form each p units of time).
    Let Sv a classical system of reference, in translation on Ox at v speed .

    a) Relatively to Sv , what is the vibratory period of (Ov) ?

    b) Relatively to Sv , what is the distance between two consecutive permanent points of (Ov) ? (Motionless points relatively to Sv).

5. Find all linear transformations of x and t, transforming any transversal wave of the elastic band in an other wave on this same elastic band.
    For each transformation found, study the behavior of  "permanent" points, study the behavior of vibratory period.

    Remark: for waves in an three dimensional elastic medium , alone Lorentz= transformations are possible.

Exercise 5.

Let systems S, S₁, S₂, S₃ . S and S₁ are classical axes with universal times.
S₁ is sliding on S with an uniform acceleration.
The equation of the origin O1 of S₁  is  x = t.t , relatively to S .
S₂ and S₃ are not classical systems.

One passes from S to S₂ by Lorentz' transformations of constants v, c, with abs(v)<c .
One passes from S₁ to S₃ by Lorentz' transformations of constants v', c' , with abs(v')<c'.
c and c' are different.

All this is perfectly coherent.
These four systems coexist, they can be physically constructed.
S₂ is for example a vibrating cord in S, cord on which, is propagating at v speed, a stationary referencial wave (see the animation on waves).
Speed of a free wave being c on this cord.
S₃ is another vibrating cord. On this cord is propagating another stationary referencial wave at v' speed .
Speed of a free wave being c' on this cord.
And more, S₃ is freely falling (relatively to S) in a uniform gravity field.

Let E an event (explosion for example).
Let (x, t); (x1, t1); (x2, t2); (x3, t3) the respective coordinates of E, relatively to S, S₁ , S₂ , S₃ .

1. Express (xi, ti) in function of (x, t).
    Express (x3, t3) from (x2, t2).

2. Let a point M, moving in S with the equation x = k t.
    Determine its speed in S₃ .
    Discuss the result in function of t .

3. Let a R ruler of (algebraic) length L, sliding on S.
    Its origin has for equation  x = k t , relatively to S , its extremity has for equation x = k t + L , relatively to S .

    a) Determine the speed of the extremity of  ruler , relatively to S_{3 .}

    b) Determine in function of .... (to you to specify), its length (noted L3) , relatively to S₃ .

Answers are gived on the french pages.