Energy - Dynamics and Relativity - page 71

p

′ 1 C

p 1 C

q C

′ 2 C

p 2 C p

Figure 3.9 Momentum diagram for an elastic ( Q = 0) collision.

p 2 c lie along a different axis to p 1 c and p 2 c , but it cannot alter the lengths of the
vectors (see Figure 3.9). The angle between these two axes defines the scattering
angle θ c . This type of scattering, in which the kinetic energy is conserved, is known
as elastic scattering.
Frequently we want to consider collisions, not between point particles, but
between objects with some internal structure. For such objects it is again possible
that we observe elastic scattering as described above. Elastic scattering occurs if
the internal structure of the colliding bodies remains unchanged by the collision
process. This happens, to a good approximation, when we collide resilient objects
such as glass or steel marbles. In general however, energy may be absorbed or
released by rearrangement of the components of the bodies and such collisions are
referred to as inelastic. We define the energy released by internal rearrangements
as the Q -value:

K − K c − Q = K = K c . (3.46)

Typically, for macroscopic bodies the Q -value is negative, i.e. the bodies absorb
energy when their internal structure changes. For example, when we drop a blob
of plasticine on the floor kinetic energy goes into deformation of the blob resulting
in a negative Q . Explosive collisions (i.e. those with Q> 0) occur when the
collision causes the release of stored internal energy; the rather contrived case of
the collision between two set mousetraps would be one example. Collisions with
Q> 0 are more common on the microscopic level, e.g. electromagnetic energy
stored within molecules can be released in exothermic chemical reactions.
Momentum diagrams for inelastic collisions are shown in Figure 3.10. Notice that
momentum conservation still applies since there are no external forces. However the
initial and final kinetic energies are now different, resulting in different magnitudes
for the initial and final momentum vectors. For Q> 0 the momentum vectors
increase in magnitude whereas for Q< 0 they decrease.

Example 3.3.1 Consider a collision between two cars each of mass m on a linear
air-track. The first car is initially travelling to the right with a speed v when it