Page 326 - Dynamics of Mechanical Systems

0593_C09_fm Page 307 Monday, May 6, 2002 2:50 PM

Principles of Impulse and Momentum 307

FIGURE 9.11.1 FIGURE 9.11.2
Colliding particles with curvilinear motion. Close view of oblique impact.

FIGURE 9.11.3
Colliding spheres.
contact, and let N be a line normal to T and passing through the contact point. We will
consider the collision to be direct impact in the N direction. That is, the linear momentum
of the system (A and B together) in the N direction is conserved. Also, in the N direction
the particles collide and rebound according to the impact rule of Eq. (9.10.4). In directions
parallel to the tangent plane, the momenta of the individual particles are conserved.
We can illustrate these ideas by considering the collision of two spheres (such as billiard
balls) as in Figure 9.11.3. Let the balls A and B be moving in the same plane on intersecting
lines. Let their velocities just before impact be v and v as in Figure 9.11.3a. Let the
B
A
collision be modeled as in Figure 9.11.3b, where n and t are unit vectors normal and
tangent to the plane of impact. Let v and v be expressed in terms of n and t as:
A
B
v = v A n v A t and v = v B n v B t (9.11.1)
+
+
A n t B n t
Similarly, let the velocities of A and B after impact be ˆ v and ˆ v , and let ˆ v and ˆ v be
A B A B
expressed as:
ˆ v = v ˆ n v+ ˆ t and ˆ v = v ˆ n v+ ˆ t (9.11.2)
A
B
A
B
A n t B n t
Let the masses of A and B be m and m . Then, conservation of linear momentum of
A
B
the system of both particles in the N direction produces the equations:
m v + mv = m v + mv ˆ B (9.11.3)
B
A
A
ˆ
An Bn An Bn