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1
Recapitulate
• We discussed EnergyMomentum Four Vector.
• In the process we gave
the new revolutionary
definition of energy.
• We gave new definition
of momentum.
Prof. Shiva Prasad, Department of Physics, IIT Bombay
2
• We gave relationships
between energy and
momentum for a single
particle.
• We discussed the
concept of zero rest
mass particle.
Prof. Shiva Prasad, Department of Physics, IIT Bombay
3
The Old Example
Completely inelastic
collision in S frame.
Before collision
m 0.6c 0.6c m
After collision
2m
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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In S before
collision
px1  mo uu  1.25  0.6moc
 0.75moc
px 2  mo uu  1.25  0.6moc
 0.75moc
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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p
k
xkI
0
Clearly the final
momentum is also zero
as the speed of the
combined particle after
collision is zero.
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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But now energy must
also be conserved, in
this frame as per our
new law, unlike before.
E1  mo uc  1.25moc
2
2
E2  mo uc 2  1.25moc 2
E
k
kI
 2.5moc
2
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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Hence the final energy
should also be same, even
though the speed of
combined particle is zero.
2
EF  2.5moc  Moc
2
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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Hence the rest mass of
the combined particle
has increased.
Mo  2.5mo
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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In S’ Frame
The velocity of the first
particle before collision.
0.6c  0.6c
u1 x 
0
1  0.36
u1 y  0
u1 z  0
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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The momentum of the
first particle before
collision is, therefore,
zero.
Prof. Shiva Prasad, Department of Physics, IIT Bombay
11
The velocity of the second
particle before collision.
0.6c  0.6c
1.2
u2 x 

c
1  0.36
1.36
u2 y  0
u2 z  0
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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The momentum of the
second particle before
collision is thus.
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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p2 x  
1.2
1.36
2
 1.2 
1

 1.36 
2.125  1.2

m0c
1.36
 1.875m0c
m0c
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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p
k
xkF
 1.875m0c
Prof. Shiva Prasad, Department of Physics, IIT Bombay
15
The velocity of the
combined particle after
collision.
0  0.6c
 
ufx
 0.6c
0  0.6c
1
2
c
 0
ufy
ufz  0
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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The momentum of the
combined particle before
collision is thus.
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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px  1.25  0.6M0c
 1.25  0.6  2.5m0c
 1.875m0c
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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Now let us check whether
energy is also conserved
in this frame.
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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Initial Energy in S’
 Ek  moc 
2
k
 3.125moc
1
 1.2 
1

 1.36 
2
moc 2
2
Prof. Shiva Prasad, Department of Physics, IIT Bombay
20
Final Energy in S’
 Ek 
k
1
1  0.6 
 1.25  2.5moc
2
Moc
2
2
 3.125moc 2
Prof. Shiva Prasad, Department of Physics, IIT Bombay
21
Another Example of
Inelastic collision
A particle of rest mass mo and
kinetic energy 6moc2 strikes and
sticks to an identical particle at rest.
what is the rest mass and speed of
the resultant particle?
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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Initial Energy
2
2
2
E  moc  6moc  moc  8moc
2
Initial Momentum
p
7moc 
2
2
2
2
 mo c 
2 2
2
E  p c  mo c
48moc
4
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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After the collision
There exists a single
particle with E=8moc2
and p= 48moc . We,
therefore, get
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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2
4
2
4
2
64mo c  48mo c  Mo c
4
Mo  4mo
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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Final Speed
E   u Moc  8moc  2Moc
2
2
2
u  2
p   u Mou
48moc  8mou
3
u
c
2
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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Alternately
1
2
u2
1 2
c
2
u
1
1 2 
4
c
3
u
c
2
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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Summary
• We discussed two examples of
what was classically known as
completely in elastic collision.
• We saw how both Energy and
Momentum are conserved in
these collisions.
Prof. Shiva Prasad, Department of Physics, IIT Bombay
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