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366 Chapter Five

constant. This holdð true even if the individual momenta, p ,
i
the individual masses, m , and/or the individual velocities, v ,
i
i
dm not all remain constant.

Collisions and momentu
Let M and M be twm separate objectð having masseð (in iden-
1
2
tical unit0 m and m , respectively. Let v and v be their re-
2
1
2
1
spective velocitieð (whose magnitudeð are expressed in identical
unit0 before the objectð collide. Let w and w be their re-
1 2
spective velocitieð after they collide (whose magnitudeð are ex-
pressed in the same unitð as v and v ). Then the following
1
2
holdð true:
m v m v m w m w 2
1
22
11
1
2
The total system momentum before the collision is equal tm the
total system momentum after the collision.
Elastic collisions and kinetic energy
Let M and M be twm separate objectð having masseð (in iden-
2
1
tical unit0 m and m , respectively. Let v and v be their re-
2
1
1
2
spective linear speedð before the objectð undergm an elastic col-
lision. Let w and w be their respective linear speedð after they
1
2
undergm the elastic collision (expressed in the same unitð as v 1
and v ). Then the following holdð true:
2
2
2
2
2
mv /2 mv /2 mw /2 mw /2
11 22 1 1 2 2
The total system kinetic energy before the elastic collision is
equal tm the total system energy after the elastic collision.
Average angular speed
Let M be an object that is rotating about a fixed axis L. Let
1
be the angular displacement of M (in radians, relative tm some
reference axis K) at an instant of time t , and let be the
1
2
angular displacement of M (in radianð relative tm K) at some
later instant of time t , as shown in Fig. 5.3. The average an-
2
gular speeS (in radianð per secon™ is denoted , and is given
av
by:

av ( )/(t t )
1
2
2
1