40
5 The Wave Equation and the Third Axiom
v
1 =
(m 1 − m 2 )v 1 + 2m 2 v 2
m 1 + m 2
,
v
2 =
2m 1 v 1 + (m 2 − m 1 )v 2
m 1 + m 2
.
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(51)
That these solutions are correct can be easily proven experimentally , e.g. letting
two steel ball bearings collide. In the extreme case where one of the masses has an
infinitely large mass, m 2 −→ ∞, an elastic reflection against a wall is the result,
v
1 = −v 1 and v
2 = v 2 = 0.
However, Eq. (51) are not the entire truth, because they result out of the solutions
(50). Equation (50) however was a result of Eq. (49), where we divided by either
(v
1 − v 1 ) or (v
2 − v 2 ) . We had to presume that these quantities were not zero, which
lead us to lose a complete class of solutions. As one can immediately see out of (49),
further solutions differing from (51) are
v
1 = v 1 ,
v
2 = v 2 .
(52)
According to (52), the particles m 1 and m 2 maintain their velocities without having
influenced each other. We could also state that they pass through each other, e.g.
the particle passes through the wall instead of reflecting from it. This second class
of solutions can also be experimentally proven and understood using Newtonian
mechanics. For example, it is not necessary that the physical particles belonging to
the two masses m 1 and m 2 on the x-axis are in fact positioned on the x-axis. Only their
geometrical centres of inertia have to be positioned there. These centres of inertia
are pure mathematical quantities. Take for example a mass m 1 , a cube where eight
(neutral) particles with the mass m 1 /8 are positioned at its corners. A second mass
m 2 with only one mass particle is positioned on the x-axis. The following ‘elastic
collision’ between the cube m 1 and the mass m 2 can be described for all, but finitely
many positions of the cube by the solution (52) and not by (51), as shown in Fig. 5.1.
Fig. 5.1 Collision of two particles with the masses m 1 and m 2 . In the illustration, it is presumed
that the total mass m 1 is divided up equally into eight particles and distributed at the corners of the
cube. The concentrated mass m 2 therefore experiences no interactions for virtually all positions of
the cube with its m 1 /8 masses, when a ‘collision’ takes place
5 The Wave Equation and the Third Axiom
v
1 =
(m 1 − m 2 )v 1 + 2m 2 v 2
m 1 + m 2
,
v
2 =
2m 1 v 1 + (m 2 − m 1 )v 2
m 1 + m 2
.
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(51)
That these solutions are correct can be easily proven experimentally , e.g. letting
two steel ball bearings collide. In the extreme case where one of the masses has an
infinitely large mass, m 2 −→ ∞, an elastic reflection against a wall is the result,
v
1 = −v 1 and v
2 = v 2 = 0.
However, Eq. (51) are not the entire truth, because they result out of the solutions
(50). Equation (50) however was a result of Eq. (49), where we divided by either
(v
1 − v 1 ) or (v
2 − v 2 ) . We had to presume that these quantities were not zero, which
lead us to lose a complete class of solutions. As one can immediately see out of (49),
further solutions differing from (51) are
v
1 = v 1 ,
v
2 = v 2 .
(52)
According to (52), the particles m 1 and m 2 maintain their velocities without having
influenced each other. We could also state that they pass through each other, e.g.
the particle passes through the wall instead of reflecting from it. This second class
of solutions can also be experimentally proven and understood using Newtonian
mechanics. For example, it is not necessary that the physical particles belonging to
the two masses m 1 and m 2 on the x-axis are in fact positioned on the x-axis. Only their
geometrical centres of inertia have to be positioned there. These centres of inertia
are pure mathematical quantities. Take for example a mass m 1 , a cube where eight
(neutral) particles with the mass m 1 /8 are positioned at its corners. A second mass
m 2 with only one mass particle is positioned on the x-axis. The following ‘elastic
collision’ between the cube m 1 and the mass m 2 can be described for all, but finitely
many positions of the cube by the solution (52) and not by (51), as shown in Fig. 5.1.
Fig. 5.1 Collision of two particles with the masses m 1 and m 2 . In the illustration, it is presumed
that the total mass m 1 is divided up equally into eight particles and distributed at the corners of the
cube. The concentrated mass m 2 therefore experiences no interactions for virtually all positions of
the cube with its m 1 /8 masses, when a ‘collision’ takes place