Page 323 - Dynamics of Mechanical Systems

0593_C09_fm Page 304 Monday, May 6, 2002 2:50 PM

304 Dynamics of Mechanical Systems

FIGURE 9.10.2 FIGURE 9.10.3
Colliding particles. Colliding particles.
*
*
*
Because v is finite, x may be made arbitrarily small by making t small. That is, for very
short impact time intervals, the displacement of P remains essentially unchanged.
Next, consider two particles A and B moving toward each other along a common line
as in Figure 9.10.2. Let the relative speed of approach be v. Let the particles collide, rebound,
ˆ v
and separate. Let the relative speed of separation be . Then, from our assumption about
the impact, we have:
ˆ v = ev (9.10.4)

where e, called the coefficient of restitution, has values between zero and one (0 ≤ e ≤ 1).
When e is zero, the separation speed is zero, and the particles stay together after impact.
ˆ v
Such collisions are said to be plastic. When e is one, the separation speed is the same as
the approach speed (that is, = v). This is called an elastic collision.
ˆ v
Equation (9.10.4) may be used with the principle of conservation of momentum to study
a variety of collision configurations. To illustrate this usage, consider first two colliding
particles A and B, each having the same mass m. Let B initially be at rest, and let A
approach B with speed v as in Figure 9.10.3. Then, from Eq. (9.10.4), the separation speed
ˆ v of the particles is ev. That is, after collision,

ˆ v − ˆ v = ev (9.10.5)
A B

Also, because the impact generates only internal forces between the particles, the overall
momentum of the system remains unchanged. That is,

mv = mv + mv ˆ (9.10.6)
ˆ
A B
Solving Eqs. (9.10.5) and (9.10.6) for ˆ v and ˆ v we obtain:
A B

ˆ v =  1 + e v and ˆ v =  1 − e v (9.10.7)
B  2  A  2 

Observe in Eq. (9.10.7) that if e is zero (plastic collision) ˆ v and ˆ v are equal with value
A B
v/2. If e is one, (elastic collision) then ˆ v is zero and ˆ v is v. In this latter case, the
A B
momentum of A is said to be transferred to B.
As a generalization of this example, consider two particles A and B with masses m and
A
m and with each particle moving on the same line before impact. Let the pre-impact
B
speeds be v and v . Let the post-impact speeds be ˆ v and ˆ v (see Figure 9.10.4). As before,
A B A B
we can determine ˆ v and ˆ v using Eq. (9.10.4) and the principle of conservation of linear
A B
momentum. From Eq. 9.10.4, we have:
e v − )
ˆ v − ˆ v = ( v (9.10.8)
B A A B