A Classical Approach of Newtonian Mechanics

6 CONSERVATION OF MOMENTUM 6.6 Collisions in 1 - dimension

is trivially satisfied, because both the left- and right-hand sides are zero. Inciden-

tally, this result is valid for both elastic and inelastic collisions.

The^ centre^ of^ mass^ kinetic^ energy^ conservation^ equation^ takes^ the^ form^

p i1 J^2 pJ^2 pJ^2 pJ^2

+^ i2^ =^

f1 + f2. (6.48)

2 m 1 2 m 2 2 m 1 2 m 2

Note, incidentally, that if energy and momentum are conserved in the laboratory

frame then they must also be conserved in the centre of mass frame. A compari-

son of Eqs. (6.45), (6.46), and (6.48) yields

(vi2 − vi1) = −(vf2 − vf1). (6.49)

In other words, the relative velocities of the colliding objects are equal and opposite

before and after the collision. This is true in all frames of reference, since relative

velocities are frame invariant. Note, however, that this result only applies to fully

elastic collisions.

Equations (6.38) and (6.49) can be combined to give the following pair of

equations which fully specify the final velocities (in the laboratory frame) of two

objects which collide elastically, given their initial velocities:

v =

(m 1 − m 2 )

v

(^) + 2 m^2 v
, (6.50)
f1 (^) m
1 +^ m 2
i1
m 1 + m 2 i2
v = 2 m^1 v
—
(m 1 − m 2 )
v
(^). (6.51)
f2 (^) m
1 +^ m 2
i1
m 1 + m 2 i2
Let us, now, consider some special cases. Suppose that two equal mass objects
collide elastically. If m 1 = m 2 then Eqs. (6.50) and (6.51) yield
vf1 = vi2, (6.52)
vf2 = vi1. (6.53)
In other words, the two objects simply exchange velocities when they collide. For
instance, if the second object is stationary and the first object strikes it head-on
with velocity v then the first object is brought to a halt whereas the second object
moves off with velocity v. It is possible to reproduce this effect in pool by striking