Page 327 - Dynamics of Mechanical Systems

0593_C09_fm Page 308 Monday, May 6, 2002 2:50 PM

308 Dynamics of Mechanical Systems

In the T direction, the momenta of each particle are conserved, producing the equations:

mv = mv ˆ A (9.11.4)
A
At At
and

mv = mv ˆ B (9.11.5)
B
Bt Bt
Finally, the impact–rebound condition of Eq. (9.10.4) in the N direction produces the
equation:

e v − )
ˆ v − ˆ v = − ( A v B (9.11.6)
A
A
n n n n
Given v and v , Eqs. (9.11.3) to (9.11.6) are four equations for the four post-impact
A
B
B
velocity components: ˆ , ˆ , ˆ ,v A v A v B and v ˆ . Because Eqs. (9.11.4) and (9.11.5) are uncoupled,
n t n t
we can readily solve for these post-impact components. The results are:
ˆ v = v A
A
t t
ˆ v = v B
B
t t
)[
A
B
A
A
B
ˆ v = (1 M m v + m v − em v + em v ] (9.11.7)
n an Bn Bn b n
)[
A
B
B
A
B
ˆ v = (1 M em v − em v + m v + m v ]
n An An An Bn
where M is defined as:
M = m + m (9.11.8)
A B
For a numerical illustration of the application of these equations, consider the collision
of two billiard balls moving prior to impact as in Figure 9.11.4. Let the balls have identical
masses and radii, and let the collision be nearly elastic with a restitution coefficient of
0.95. From Figure 9.11.4a, the pre-impact speed components are:
A
B
B
A
.
v = 14 14cm sec , v = 14 14cm sec , v = −12 99cm sec , v = 7 5cm sec (9.11.9)
.
.
.
t
t
n
n
Then, from Eq. (9.11.7), the post-impact speed components are:
B
A
A
B
.5
ˆ v =−12 .31cm sec , ˆ v = 14 .14cm sec , ˆ v = 1 .12cm sec , ˆ v = 7 cm sec (9.11.10)
n t n t
The post-impact motion of the balls may then be depicted as in Figure 9.11.5.