68 5. IMPULSE, MOMENTUM, AND IMPACT OF PARTICLES
Similarly, by considering particle B, one can show
e D
v
0
B
u
u v
B
: (5.13)
From Equations (5.12) and (5.13) the unknown velocity u is to be eliminated.
From Equation (5.12), one obtains
e
.
v
A
u
/
D u v
0
A
:
Rearranging terms to obtain u,
u D
ev
A
C v
0
A
1 C e
: (5.14)
From Equation (5.13), one has
u D
ev
B
C v
0
B
1 C e
: (5.15)
Equating (5.14) to (5.15), it gives
ev
A
C v
0
A
D ev
B
C v
0
B
:
Rearranging terms, one has
e D
v
0
B
v
0
A
v
A
v
B
: (5.16)
Equation (5.16) expresses the coefficient of restitution as a ratio of the relative velocity after
impact to that before impact of the two particles. Applying Equation (5.7) and (5.16) the two
unknown velocities after impact, v
0
A
and v
0
B
, can be solved.
Remarks:
(a) For perfectly plastic impact, e D 0:
(b) For perfectly elastic impact, e D 1: us, in this case, the total energy as well as the total
momentum of the two particles is conserved. at is, the kinetic energy of the two particles
is conserved
1
2
m
A
v
2
A
C
1
2
m
B
v
2
B
D
1
2
m
A
v
0
A
2
C
1
2
m
B
v
0
B
2
: (5.17)
5.3.2 OBLIQUE CENTRAL IMPACT
Now, consider two colliding particles whose velocities are not directed along the line of impact,
as shown in Figure 5.2b. is is the case known as oblique central impact. Suppose the velocities
*
v
0
A
and
*
v
0
B
after impact are in the directions shown in Figure 5.5. ese unknown vector quan-
tities constitute four unknowns, two for unknown magnitudes and two for unknown angles or
directions. eir solution requires four independent equations.
5.3. IMPULSIVE MOTION AND IMPACT 69
Assume coordinate n is along the line of impact and coordinate t, perpendicular to n, is
the common tangent. If the surfaces of the particles are perfectly smooth and frictionless as well
as no external force applied to the particles, the only impulses acted on the particles during the
impact are the internal forces directed along the line of impact. at is, the internal forces are
along the n-coordinate in Figure 5.5. is means that the impact of the two particles has the
following three situations.
⃑'
A
⃑'
B
⃑
A
⃑
B
Line of
Impact
t
n
A
B
Figure 5.5: Oblique central impact of two particles.
First, the component along the t-axis of the momentum of each particle is conserved. at
is, the t component of the velocity of each particle remains unchanged. is means that
*
v
A
t
D
*
v
0
A
t
;
(5.18a)
*
v
B
t
D
*
v
0
B
t
(5.18b)
in which the subscript t denotes the t component of the velocity.
Second, the component along the n-axis of the two momenta of the two particles is con-
served. is enables one to write
m
A
*
v
A
n
C m
B
*
v
B
n
D m
A
*
v
0
A
n
C m
B
*
v
0
B
n
: (5.19)
ird, the component along the n-axis of the relative velocity of the two particles after
impact can be obtained by multiplying the coefficient of restitution and the relative velocity
before impact. is is similar to that for the direct central impact. is gives
*
v
0
B
n
*
v
0
A
n
D e
*
v
A
n
*
v
B
n
: (5.20)
e four independent equations, Equations (5.18a), (5.18b), (5.19), and (5.20), can be ap-
plied to solve for the four unknowns (two unknown magnitudes and two unknown phases
or angles) of the velocities
*
v
0
A
and
*
v
0
B
of the two particles after impact.
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