Page 359 - Dynamics of Mechanical Systems

0593_C10_fm Page 340 Monday, May 6, 2002 2:57 PM

340 Dynamics of Mechanical Systems

Next, during impact, the momentum is conserved. That is,

mv = ( m + )
1 b 1 m v a (10.10.7)
2
where v is the speed of the first vehicle just before impact (see Eq. (9.7.1)).
b
From Eq. (10.10.2), the work W done by friction forces on the first vehicle as it slides a
distance d before the collision is:
1
W =−µ m gd (10.10.8)
1 1

If v is the speed of the first vehicle when skidding begins, the change in kinetic energy
0
of the vehicle from the beginning of skidding until the collision is:
∆K = ( ) m v −( ) 2
2
12
1 b 12 m v (10.10.9)
10
The work–energy principle then gives:
W = ∆ K or − µ m gd = ( ) m v −( ) m v 2 (10.10.10)
2
12
12
1 1 1 b 10
Finally, using Eqs. (10.10.6) and (10.10.7), we can solve Eq. (10.10.10) for v 2 :
0
1 [
2 (
v = v + 2µ gd = 1+ m m )] 2 v + 2µ gd 1
2
2
2
b
0
1
a
(10.10.11)
2 (
= [ 1+ m m )] 2 2µ gd + 2µ gd 1
1
2
Observe from Eq. (10.10.7) that the speed v of the first vehicle just after the collision
a
with the second vehicle is reduced by the factor [m /(m + m )]. That is, the change in
2
1
1
speed ∆v is:
b [
∆v = v − v = − m ( m + )] v (10.10.12)
m
a 2 1 2 b
Observe further that because the velocity changes during the impact the kinetic energy
also changes. That is, even though the momentum is conserved, the kinetic energy is not
conserved. Indeed, the change in kinetic energy ∆K just before and just after the impact is:
2
∆K = ( ) m v −( ) m v =−( ) m   2 m + 2 m m + m 2 3   v 2 (10.10.13)
2
2
2
1
1
12
12
12
1
1
1
b
a
  ( m + ) 2   b
m
2
1
In actual accidents, the vehicles do not usually leave uniform skid marks from all four
wheels. Also, collisions are not usually perfectly plastic nor do the vehicles always move
in a straight line on a level surface. However, with minor modifications, the work–energy
and momentum conservation principles may still be used to obtain reasonable estimates
of vehicle speeds at various stages of an accident. The details of these modifications and
the corresponding application of the principles are beyond the scope of this text; the reader
is referred to the references for further information.