A Classical Approach of Newtonian Mechanics

6 CONSERVATION OF MOMENTUM 6.6 Collisions in 1 - dimension

y vf2

m 2

m 1

vi1

 2

m 2 

1

m 1

i1

x

vf1

Figure 55: A collision in 2 - dimensions.

two momenta must also be equal (since the two objects stick together). The only

way in which this is possible is if the two objects remain stationary in the centre

of mass frame after the collision. Hence, the two objects move with the centre of

mass velocity in the laboratory frame.

Suppose that the second object is initially at rest (i.e., vi2 = 0 ). In this special
case, the common final velocity of the two objects is

v = m^1 v.^ (6.59)^

f (^) m
1 +^ m 2
i1
Note that the first object is slowed down by the collision. The fractional loss in
kinetic energy of the system due to the collision is given by
Ki − Kf m 1 v 2 − (m 1 + m 2 ) v 2 m 2
f = =
Ki
i1 f
m 1 v 2
=. (6.60)
m 1 + m 2
The loss in kinetic energy is small if the (initially) stationary object is much lighter
than the moving object (i.e., if m 2 m 1 ), and almost 100 % if the moving object
is much lighter than the stationary one (i.e., if m 2 m 1 ). Of course, the lost
kinetic energy of the system is converted into some other form of energy: e.g.,
heat energy.