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, S _{1}, S_{2}, S_{3} **.

The equation of the origin O1 of

One passes from **S** to **S _{2}** by Lorentz' transformations
of constants v, c, with abs(v)<c .

One passes from

c and c' are different.

All this is perfectly coherent.

These four systems coexist, they can be physically constructed.
**S _{2}** is for example a vibrating cord in

Speed of a free wave being c on this cord.

Speed of a free wave being c' on this cord.

And more,

Let E an event (explosion for example).

Let (x, t); (x1, t1); (x2, t2); (x3, t3) the respective coordinates
of E, relatively to **S**, **S _{1}** ,

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 **S _{3}** .

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 **S _{3}** .

Answers are gived on the french pages.