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Lecture 7
COLLISIONS
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Lecture 7
COLLISIONS
1 2-d elastic collision in Lab Frame
2 Collision in CM reference frame
3 Advantages of CM Frame
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The Large Hadron Collider (LHC)
Collisions 2/16
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The Large Hadron Collider (LHC)
The aim of the exercise:
To smash protons moving at 99.999999% of the speed oflight into each other and so recreate conditions a fraction of a
second after the big bang. The LHC experiments try and
work out what happened.
Collisions 2/16
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The Large Hadron Collider (LHC)
The aim of the exercise:
To smash protons moving at 99.999999% of the speed oflight into each other and so recreate conditions a fraction of a
second after the big bang. The LHC experiments try and
work out what happened.
proton-proton collisions at 7 TeV each (14 TeV in CM frame!)
Collisions 2/16
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The Large Hadron Collider (LHC)
The aim of the exercise:
To smash protons moving at 99.999999% of the speed oflight into each other and so recreate conditions a fraction of a
second after the big bang. The LHC experiments try and
work out what happened.
proton-proton collisions at 7 TeV each (14 TeV in CM frame!)
2808 bunches of 1.15 1011 protons per bunch
Collisions 2/16
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The Large Hadron Collider (LHC)
The aim of the exercise:
To smash protons moving at 99.999999% of the speed oflight into each other and so recreate conditions a fraction of a
second after the big bang. The LHC experiments try and
work out what happened.
proton-proton collisions at 7 TeV each (14 TeV in CM frame!)
2808 bunches of 1.15 1011 protons per bunch
to find The Higgs Boson New particles
Collisions 2/16
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Collisions
Standard experimental method to discover the inner constituents
of matter
To find out the nature of sub-atomic forces
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Collisions
Ernst Rutherford (1910): -particle-gold collisions
discovered NucleusCollisions 5/16
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Collisions
James Chadwick (1932): -particle-Beryllium collisions
discovered Neutron
Collisions 6/16
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Laws of Collisions
Characterizing a Collision
Collisions 7/16
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Laws of Collisions
Characterizing a Collision
1 A very short-duration interaction
Collisions 7/16
C
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Laws of Collisions
Characterizing a Collision
1 A very short-duration interaction2 External forces are ignorable
Collisions 7/16
L f C lli i
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Laws of Collisions
Characterizing a Collision
1 A very short-duration interaction2 External forces are ignorable
Conservations that help solve the problem:
Collisions 7/16
L f C lli i
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Laws of Collisions
Characterizing a Collision
1 A very short-duration interaction2 External forces are ignorable
Conservations that help solve the problem:
1 Linear momentum is conserved:P = 0
Collisions 7/16
L f C lli i
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Laws of Collisions
Characterizing a Collision
1 A very short-duration interaction2 External forces are ignorable
Conservations that help solve the problem:
1 Linear momentum is conserved:P = 0
2 Kinetic Energy:
Collisions 7/16
La s of Collisions
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Laws of Collisions
Characterizing a Collision
1 A very short-duration interaction2 External forces are ignorable
Conservations that help solve the problem:
1 Linear momentum is conserved:P = 0
2 Kinetic Energy:
Elastic: KE conserved
Collisions 7/16
Laws of Collisions
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Laws of Collisions
Characterizing a Collision
1 A very short-duration interaction2 External forces are ignorable
Conservations that help solve the problem:
1 Linear momentum is conserved:P = 0
2 Kinetic Energy:
Elastic: KE conservedInelastic: KE = Q
Collisions 7/16
Laws of Collisions
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Laws of Collisions
Characterizing a Collision
1 A very short-duration interaction2 External forces are ignorable
Conservations that help solve the problem:
1 Linear momentum is conserved:P = 0
2 Kinetic Energy:
Elastic: KE conservedInelastic: KE = Q
Q < 0: KE converted to energy released (Explosions)
Collisions 7/16
Laws of Collisions
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Laws of Collisions
Characterizing a Collision
1 A very short-duration interaction2 External forces are ignorable
Conservations that help solve the problem:
1 Linear momentum is conserved:P = 0
2 Kinetic Energy:
Elastic: KE conservedInelastic: KE = Q
Q < 0: KE converted to energy released (Explosions)Q > 0 KE is absorbed
Collisions 7/16
2 d elastic collision in Lab Frame
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2-d elastic collision in Lab Frame
Collisions 2-d elastic collision in Lab Frame 8/16
2-d elastic collision in Lab Frame
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2-d elastic collision in Lab Frame
Conservation Equations:
Collisions 2-d elastic collision in Lab Frame 8/16
2-d elastic collision in Lab Frame
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2-d elastic collision in Lab Frame
Conservation Equations:
px: m1v1 = m1v
1 cos 1 + m2v
2 cos 2
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2-d elastic collision in Lab Frame
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2-d elastic collision in Lab Frame
Conservation Equations:
px: m1v1 = m1v
1 cos 1 + m2v
2 cos 2
py: m1v
1 sin 1 = m2v
2 sin 2
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2-d elastic collision in Lab Frame
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2 d elastic collision in Lab Frame
Conservation Equations:
px: m1v1 = m1v
1 cos 1 + m2v
2 cos 2
py: m1v
1 sin 1 = m2v
2 sin 2
KE:1
2m1v
2
1 =1
2m1v
2
1 +1
2m2v
2
2
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2-d elastic collision in Lab Frame
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2 d elastic collision in Lab Frame
Conservation Equations:
px: m1v1 = m1v
1 cos 1 + m2v
2 cos 2
py: m1v
1 sin 1 = m2v
2 sin 2
KE:1
2m1v
2
1 =1
2m1v
2
1 +1
2m2v
2
2
4 eqns, 3 unknowns!
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2-d elastic collision in Lab Frame
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2 d elastic collision in Lab Frame
Conservation Equations:
px: m1v1 = m1v
1 cos 1 + m2v
2 cos 2
py: m1v
1 sin 1 = m2v
2 sin 2
KE:1
2m1v
2
1 =1
2m1v
2
1 +1
2m2v
2
2
4 eqns, 3 unknowns!
One final stage parameter has to be expt. measured! e.g 1.
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2-d elastic collision in Lab Frame
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2 d elastic collision in Lab Frame
Conservation Equations:
px: m1v1 = m1v1 cos 1 + m2v
2 cos 2
py: m1v
1 sin 1 = m2v
2 sin 2
KE:1
2m1v
2
1 =1
2m1v
2
1 +1
2m2v
2
2
4 eqns, 3 unknowns!
One final stage parameter has to be expt. measured! e.g 1.
Solving,
v1 =m1v1 cos 1
m1 + m2 v1
m2
1cos2 1
(m1 + m2)2
m1 m2m1 + m2
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2-d elastic collision in Lab Frame
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Analysis:
1 Ifm
1
> m2,
1 has a max. limiting value:
Collisions 2-d elastic collision in Lab Frame 9/16
2-d elastic collision in Lab Frame
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Analysis:
1 If m1 > m2, 1 has a max. limiting value: cos2
max= 1
m22
m21
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2-d elastic collision in Lab Frame
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Analysis:
1 If m1 > m2, 1 has a max. limiting value: cos2
max= 1
m22
m21 For 1 < max two possible values of v
1
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2-d elastic collision in Lab Frame
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Analysis:
1 If m1 > m2, 1 has a max. limiting value: cos2
max= 1
m22
m21 For 1 < max two possible values of v
1
m1 m2, max 0
Collisions 2-d elastic collision in Lab Frame 9/16
2-d elastic collision in Lab Frame
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Analysis:
1 If m1 > m2, 1 has a max. limiting value: cos2
max= 1
m22
m21 For 1 < max two possible values of v
1
m1 m2, max 0
2
Head-on collisions: 1
= 0
= v1 = v1m1 m2m1 + m2
,
Collisions 2-d elastic collision in Lab Frame 9/16
2-d elastic collision in Lab Frame
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Analysis:
1 If m1 > m2, 1 has a max. limiting value: cos2 max = 1
m22
m21 For 1 < max two possible values of v
1
m1 m2, max 0
2
Head-on collisions: 1
= 0
= v1 = v1m1 m2m1 + m2
, v2 = v12m1
m1 + m2
Collisions 2-d elastic collision in Lab Frame 9/16
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2-d elastic collision in Lab Frame
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Analysis:
1 If m1 > m2, 1 has a max. limiting value: cos2 max = 1
m22
m21 For 1 < max two possible values of v
1
m1 m2, max 0
2
Head-on collisions: 1
= 0
= v1 = v1m1 m2m1 + m2
, v2 = v12m1
m1 + m2
Final KE of m2 =4 m1/m2
(1 + m1/m2 )2intial KE of m1.
i.e. measuring initial & final KE gives ratio of masses
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2-d elastic collision in Lab Frame
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Analysis:
1 If m1 > m2, 1 has a max. limiting value: cos2 max = 1
m22
m2
1
For 1 < max two possible values of v
1
m1 m2, max 0
2 Head-on collisions:1
= 0
= v1 = v1m1 m2m1 + m2
, v2 = v12m1
m1 + m2
Final KE of m2 =4 m1/m2
(1 + m1/m2 )2intial KE of m1.
i.e. measuring initial & final KE gives ratio of masses
Chadwicks discovery of neutron
Collisions 2-d elastic collision in Lab Frame 9/16
Collision in CM reference frame
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CM frame is inertial:
Collisions Collision in CM reference frame 10/16
Collision in CM reference frame
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CM frame is inertial:
RCM =m1r1 + m2r2
m1 + m2
Collisions Collision in CM reference frame 10/16
Collision in CM reference frame
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CM frame is inertial:
RCM =m1r1 + m2r2
m1 + m2
Here, VCM =m1v1
m1 + m2, constant.
Collisions Collision in CM reference frame 10/16
Collision in CM reference frame
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CM frame is inertial:
RCM =m1r1 + m2r2
m1 + m2
Here, VCM =m1v1
m1 + m2, constant.
Velocity transformations:
v1c = v1 VCM =m2
m1 + m2v1
Collisions Collision in CM reference frame 10/16
Collision in CM reference frame
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CM frame is inertial:
RCM =m1r1 + m2r2
m1 + m2
Here, VCM =m1v1
m1 + m2, constant.
Velocity transformations:
v1c = v1 VCM =m2
m1 + m2v1
v2c = v2 VCM = m1
m1 + m2v1
Collisions Collision in CM reference frame 10/16
Collision in CM reference frame
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CM frame is inertial:
RCM =m1r1 + m2r2
m1 + m2
Here, VCM =m1v1
m1 + m2, constant.
Velocity transformations:
v1c = v1 VCM =m2
m1 + m2v1
v2c = v2 VCM = m1
m1 + m2v1
v
1c = v
1
VCM
Collisions Collision in CM reference frame 10/16
Collision in CM reference frame
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CM frame is inertial:
RCM =m1r1 + m2r2
m1 + m2
Here, VCM =m1v1
m1 + m2, constant.
Velocity transformations:
v1c = v1 VCM =m2
m1 + m2v1
v2c = v2 VCM = m1
m1 + m2v1
v
1c = v
1
VCM
Collisions Collision in CM reference frame 10/16
Collision in CM reference frame
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CM frame is inertial:
RCM =m1r1 + m2r2
m1 + m2
Here, VCM =m1v1
m1 + m2, constant.
Velocity transformations:
v1c = v1 VCM =m2
m1 + m2v1
v2c = v2 VCM = m1
m1 + m2v1
v
1c =v
1 V
CM
v
2c = v
2 VCM
Collisions Collision in CM reference frame 10/16
Collision in CM reference frame
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This is a zero-momentum frame
Collisions Collision in CM reference frame 11/16
Collision in CM reference frame
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This is a zero-momentum frame
Momentum Conservation:
m1v1c = m2v2c
m1v
1c = m2v
2c
Trajectories turn through anangle .
Collisions Collision in CM reference frame 11/16
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Collision in CM reference frame
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This is a zero-momentum frame
Momentum Conservation:
m1v1c = m2v2c
m1v
1c = m2v
2c
Trajectories turn through anangle .
Magnitudes of velocities: v2c =m1m2
v1c; v
2c =m1m2
v1c
KE Conservation: v1c
= v2c
i.e speeds of the particles are the same in CM frame for an elastic
collision.
Collisions Collision in CM reference frame 11/16
Advantages of CM frame
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1 Momentum is automatically conserved
2
Only one angle to worry about
Collisions Collision in CM reference frame 12/16
Relationship between scattering angles and 1
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tan 1 =
Collisions Collision in CM reference frame 13/16
Relationship between scattering angles and 1
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tan 1 =
v1c sin
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Relationship between scattering angles and 1
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tan 1 =
v1c sin
VCM + v
1c cos
=sin
cos + VCMv1c
Collisions Collision in CM reference frame 13/16
Relationship between scattering angles and 1
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Since v1c = v1c =m2m1
VCM,
tan 1 =
v1c sin
VCM + v
1c cos
=sin
cos + VCMv1c
Collisions Collision in CM reference frame 13/16
Relationship between scattering angles and 1
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Since v1c = v1c =m2m1
VCM,
tan 1 =
v1c sin
VCM + v
1c cos
=sin
cos + VCMv1c
tan 1 =sin
cos + m1m2
Cases:
Collisions Collision in CM reference frame 13/16
Relationship between scattering angles and 1
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Since v1c = v1c =m2m1
VCM,
tan 1 =
v1c sin
VCM + v
1c cos
=sin
cos + VCMv1c
tan 1 =sin
cos + m1m2
Cases:
1 m1 < m2: No restriction on 1
Collisions Collision in CM reference frame 13/16
Relationship between scattering angles and 1
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Since v1c = v1c = m2
m1VCM,
tan 1 =
v1c sin
VCM + v
1c cos
=sin
cos + VCMv1c
tan 1 =sin
cos + m1m2
Cases:
1 m1 < m2: No restriction on 1
2 m1 = m2: 1 [0,
2]
Collisions Collision in CM reference frame 13/16
Relationship between scattering angles and 1
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Since v1c = v1c = m2
m1VCM,
tan 1 =
v1c sin
VCM + v
1c cos
=sin
cos + VCMv1c
tan 1 = sincos + m1
m2
Cases:
1 m1 < m2: No restriction on 1
2 m1 = m2: 1 [0,
2]
3 m1 > m2: 1 has a maximum limiting value
Collisions Collision in CM reference frame 13/16
Advantages of CM frame (contd)
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Energy expressions:
Collisions Advantages of CM Frame 14/16
Advantages of CM frame (contd)
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Energy expressions:
KE in lab frame expressed in terms of VCM, v1c and v2c
Collisions Advantages of CM Frame 14/16
Advantages of CM frame (contd)
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Energy expressions:
KE in lab frame expressed in terms of VCM, v1c and v2c
KE =1
2m1(v1c + VCM)
2 +1
2m2(v2c + VCM)
2
= 12
m1v2
1c +1
2m2v
2
2c
+ (m1v1c + m2v2c) VCM
+ 12
(m1 + m2)V2
CM
Collisions Advantages of CM Frame 14/16
Advantages of CM frame (contd)
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Energy expressions:
KE in lab frame expressed in terms of VCM, v1c and v2c
KE =1
2m1(v1c + VCM)
2 +1
2m2(v2c + VCM)
2
= 12
m1v2
1c +1
2m2v
2
2c KE1c + KE2c
+ (m1v1c + m2v2c) VCM
+ 12
(m1 + m2)V2
CM
Collisions Advantages of CM Frame 14/16
Advantages of CM frame (contd)
E i
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Energy expressions:
KE in lab frame expressed in terms of VCM, v1c and v2c
KE =1
2m1(v1c + VCM)
2 +1
2m2(v2c + VCM)
2
= 12
m1v2
1c +1
2m2v
2
2c KE1c + KE2c
+ (m1v1c + m2v2c) VCM 0
+ 12
(m1 + m2)V2
CM
Collisions Advantages of CM Frame 14/16
Advantages of CM frame (contd)
E i
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Energy expressions:
KE in lab frame expressed in terms of VCM, v1c and v
2c
KE =1
2m1(v1c + VCM)
2 +1
2m2(v2c + VCM)
2
= 12
m1v2
1c +1
2m2v
2
2c KE1c + KE2c
+ (m1v1c + m2v2c) VCM 0
+ 12
(m1 + m2)V2
CM KECM
Collisions Advantages of CM Frame 14/16
Advantages of CM frame (contd)
E i
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Energy expressions:
KE in lab frame expressed in terms of VCM, v1c and v
2c
KE =1
2m1(v1c + VCM)
2 +1
2m2(v2c + VCM)
2
= 12
m1v2
1c +1
2m2v
2
2c KE1c + KE2c
+ (m1v1c + m2v2c) VCM 0
+ 12
(m1 + m2)V2
CM KECM
KE = KEI + KECM
Collisions Advantages of CM Frame 14/16
Advantages of CM frame (contd)
Energ e pressions
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Energy expressions:
KE in lab frame expressed in terms of VCM, v1c and v
2c
KE =1
2m1(v1c + VCM)
2 +1
2m2(v2c + VCM)
2
= 12
m1v2
1c +1
2m2v
2
2c KE1c + KE2c
+ (m1v1c + m2v2c) VCM 0
+ 12
(m1 + m2)V2
CM KECM
KE = KEI + KECM
Theorem
KE of a system =
Collisions Advantages of CM Frame 14/16
Advantages of CM frame (contd)
Energy expressions:
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Energy expressions:
KE in lab frame expressed in terms of VCM, v1c and v
2c
KE =1
2m1(v1c + VCM)
2 +1
2m2(v2c + VCM)
2
= 12
m1v2
1c +1
2m2v
2
2c KE1c + KE2c
+ (m1v1c + m2v2c) VCM 0
+ 12
(m1 + m2)V2
CM KECM
KE = KEI + KECM
Theorem
KE of a system =internal KE (relative to CM)
Collisions Advantages of CM Frame 14/16
Advantages of CM frame (contd)
Energy expressions:
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Energy expressions:
KE in lab frame expressed in terms of VCM, v1c and v
2c
KE =1
2m1(v1c + VCM)
2 +1
2m2(v2c + VCM)
2
= 12
m1v2
1c +1
2m2v
2
2c KE1c + KE2c
+ (m1v1c + m2v2c) VCM 0
+ 12
(m1 + m2)V2
CM KECM
KE = KEI + KECM
Theorem
KE of a system = internal KE (relative to CM) +KE of CM
Collisions Advantages of CM Frame 14/16
Advantages of CM frame (contd)
Of the total initial KE of system only part is available for use:
http://find/http://goback/8/8/2019 Lec9 Collisions
71/73
Of the total initial KE of system, only part is available for use:
KEc = KEinitial KECM
=1
2
m1m2m1 + m2
(v2 v1)2
=1
2
v2rel
Collisions Advantages of CM Frame 15/16
Advantages of CM frame (contd)
Of the total initial KE of system only part is available for use:
http://find/http://goback/8/8/2019 Lec9 Collisions
72/73
Of the total initial KE of system, only part is available for use:
KEc = KEinitial KECM
=1
2
m1m2m1 + m2
(v2 v1)2
=1
2
v2rel
For an inelastic collision, KE = Q we require the masses to
have a minimum available energy for the collision to be effective:
KEc|min = Q
Collisions Advantages of CM Frame 15/16
References
LHC Homepage:
http://find/http://goback/8/8/2019 Lec9 Collisions
73/73
LHC Homepage:
http://lhc-machine-outreach.web.cern.ch/lhc-machine-outreach/
Images:
http://www.boston.com/bigpicture/2008/08/the large hadron collider.
Collisions Advantages of CM Frame 16/16
http://find/http://goback/Recommended