The initial velocity of the cue ball is
V
=
V
j
=
V
cos
α
n
+
V
sin
α
t
TangentialVelocity Invariance.
The velocity of the balls is unchanged in the
t
direction. Hence, for the
cue ball, we have
V
I
t
=
V
sin
α
NormalMomentum Conservation.
Letting
V
I
8
denote the speed of the eight ball in the
n
direction after
the impact, there follows
mV
cos
α
+
m
·
0 =
mV
I
n
+
mV
I
8
=
⇒
V
I
n
+
V
I
8
=
V
cos
α
Impact Relation.
The final principle we use is the impact relation, viz.,
V
I
8
−
V
I
n
=
e
(
V
cos
α
−
0)
=
⇒
V
I
8
−
V
I
n
=
eV
cos
α
Completing the Solution.
Subtracting the impactrelation equation from the
n
momentum equation yields
V
I
n
=
1
2
(1
−
e
)
V
cos
α
Therefore, the velocity of the cue ball after the impact is
V
I
=
1
2
(1
−
e
)
V
cos
α
n
+
V
sin
α
t
To transform to
xy
coordinates, we substitute for
n
and
t
from above and proceed as follows.
V
I
=
1
2
(1
−
e
)
V
cos
α
(
−
sin
α
i
+ cos
α
j
) +
V
sin
α
(cos
α
i
+ sin
α
j
)
=
}
−
1
2
(1
−
e
)
V
sin
α
cos
α
+
V
sin
α
cos
α
]
i
+
}
1
2
(1
−
e
)
V
cos
2
α
+
V
sin
2
α
]
j
=
}
1
2
(1 +
e
)
V
sin
α
cos
α
]
i
+
}
1
2
(1
−
e
)
V
cos
2
α
+
V
sin
2
α
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 Spring '06
 Shiflett
 Velocity, Cos, ball, Billiard ball, Holden Commodore

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