inelastic relativistic collision
A particle of mass
m
, moving at speed
v
= 4
c
/5, collides inelastically with a similar
particle at rest.
(a) What is the speed
v
C
of the composite particle?
(b) What is its mass
m
C
?
Solution by Rudy Arthur:
Call the moving particle ‘M’, and the particle at rest ‘R’ (the composite particle is defined
to be ‘C’).
The momentum of the moving particle is
M
2
2
4
3
1
mv
p
v
c
=
=
−
mc
M
2
.
(1)
And, the square of its energy is
M
2
2
2
(
)
(
)
E
mc
p c
=
+
.
(2)
The energy of the particle at rest is
R
2
(
)
E
mc
=
.
(3)
The square of the energy of the composite particle
is
C
C
C
2
2
2
(
)
(
2
)
E
m c
p c
=
+
.
(4)
By conservation of energy:
, or squaring and rearranging,
M
R
E
E
E
+
=
C
2
M
R
C
M
R
2
2
2
E E
E
E
E
=
−
−
(5)
Substituting (2) and (4) into (5):
(
)
(
)
M
R
C
C
M
2
2
2
2
2
2
2
(
)
(
)
2(
)
(
)
E E
m c
p c
mc
p c
=
+
−
+
By conservation of momentum,
C
M
p
p
=
, so this reduces to
M
R
C
2
2
2
2
2
(
)
2(
)
E E
m c
mc
=
−
Squaring again:
(
M
R
C
2
2
2
2
2

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