Page 325 - Dynamics of Mechanical Systems

0593_C09_fm Page 306 Monday, May 6, 2002 2:50 PM

306 Dynamics of Mechanical Systems

12
ˆ v = ( ) ( [ 1 + e v ) +(1 − e v ) ] (9.10.18)
B A B
Suppose further in this case that e = 1 (elastic collision). Then,

ˆ v = v and ˆ v = v (9.10.19)
A B B A

If, instead, e = 0 (plastic collision), then we have:

ˆ v = ( )( ˆ v = ( )( v + )
v
A
A 12 v + ) and B 12 A v B (9.10.20)
B
Finally, if v is zero in Eqs. (9.10.17) and (9.10.18), then ˆ v A and ˆ v B are:
B
ˆ v =  1 − e v and ˆ v =  1 + e v (9.10.21)
A  2  A B  2  A

These results are seen to be consistent with those of Eq. (9.10.7).
For still another specialization of Eqs. (9.10.10) and (9.10.11), let the mass of A greatly
exceed that of B, let the collision be elastic, and let B be initially at rest. That is,

m >> m , e = , v = 0 (9.10.22)
1
A B B
Then, ˆ v and ˆ v become:
A B
ˆ v = v and v = 2 v (9.10.23)
A A B A

Observe that the struck particle B attains a speed twice as large as that of the striking
particle A.

9.11 Oblique Impact

When colliding particles are moving along the same straight line (as in the foregoing
examples) the collision is called direct impact. Generally, however, when particles (or
bodies) collide they are not moving on the same line. Instead, they are usually moving
on distinct curves that intersect. Such collisions are called oblique impacts. It happens that
oblique impacts may be studied using the same principles we used with direct impacts.
That is, oblique impact is treated as a direct impact in the direction normal to the plane
of contact of the bodies. In directions parallel to the plane of contact, the momenta of the
individual bodies are conserved.
To develop the analysis consider the oblique impact depicted in Figure 9.11.1, where A
and B are particles moving on intersecting curves. Consider also the closer view of the
impact provided in Figure 9.11.2. Let the particles be considered to be “small” bodies.
That is, we will neglect rotational or angular momentum effects in the analysis. In
Figure 9.11.2, let T represent a plane tangent to the contacting surfaces at the point of