Physics 105A
Analytical Mechanics
Center of Mass Frame
1-D and 2-D Collisions
28 June 2016
Manuel Calderón de la Barca Sánchez
Relative velocities between frames
 We always measure velocities and momenta relative to
some inertial frame.
 Consider two frames:
u
S’
S
 Velocity of i’th particle in frame S:
28 June 2016
MCBS
vi = vi '+ u
5.6.1 Center of Mass Frame
 Assume that momentum is conserved in frame S’.
What does this imply for momentum conservation in frame S?
 Different frames will have different values of the total
momentum.
P º å mi vi
i
What if we choose a frame in which P = 0 ?
P º å mi vi in frame S, then the Center of Mass frame
 If
i
is the frame which moves at velocity:
28 June 2016
MCBS
mv
å
P
u=
º
M åm
i i
i
i
i
1-D Collisions
m
v
M
 Elastic collisions:
Conserve momentum
Conserve energy
 Can treat motion in lab frame or CM frame (or whatever
frame is most convenient).
 Example above: Before collision, mass m moves at v,
mass M is stationary. Find final velocities after collision.
28 June 2016
MCBS
m
v
M
 What is the total momentum in lab frame?
 What is the speed of the CM?
 What are the speeds of the masses in the CM frame?
Check: relative speed in CM frame should still be v.
Check: total momentum in CM frame should be 0.
– Ratio of speeds?
 What happens to total momentum after collision.
What does this mean for the total energy? For the speeds?
28 June 2016
MCBS
m
v
M
 Final lab velocities:
m-M
vm = v
m+ M
2m
vM = v
m+ M
 Interesting limits:
m=M
M>>m
M<<m
28 June 2016
MCBS
5.6.2 Kinetic Energy between CM
frame and other frames
 Kinetic energy in CM:
KCM
1
2
= å mi vi '
2 i
Relationship between Kinetic energy in CM and any other
frame moving at speed u with respect to CM frame turns out to
be quite nice:
1
2
K S = K CM + Mu
2
 Elastic collisions: P and KS
are conserved.
 Theorem 5.3: In a 1-D elastic collision, the relative velocity
of the two particles after the collision is the negative of the
relative velocity before the collision.
28 June 2016
MCBS
5.7.2 Collisions in 2-D
 A billiard ball with speed v approaches an identical
stationary ball. The balls bounce off each other elastically,
in such a way that the incoming one gets deflected by an
angle q.
What are the final speeds of the balls.
What is the angle f at which the stationary ball is deflected?
What is the angle between the final v’s of the two balls?
q
f
28 June 2016
MCBS