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10.58.
Model:
Model the balls as particles. We will use the Galilean transformation of velocities (Equation
10.44) to analyze the problem of elastic collisions. We will transform velocities from the lab frame S to a frame S
′
in which one ball is at rest. This allows us to apply Equations 10.43 to a perfectly elastic collision in S
′
. After
finding the final velocities of the balls in S
′
, we can then transform these velocities back to the lab frame S.
Visualize:
Let S
′
be the frame of the 400 g ball. Denoting masses as
1
100 g
m
=
and
2
400 g,
m
=
the initial
velocities in the S frame are
i1
()
4
.
0
m
/
s
x
v
=+
and
i2
1
.
0
m
/
s
.
x
v
Figures (a) and (b) show the beforecollision situations in frames S and S,
′
respectively. The aftercollision
velocities in S
′
are shown in figure (c). Figure (d) indicates velocities in S after they have been transformed to S
from S .
′
Solve:
In frame S,
() 4
.
0
m
/
s
x
v
=
and
() 1
.
0
m
/
s
.
x
v
=
Because S
′
is the reference frame of the 400 g ball,
1.0 m/s.
V
=
The velocities of the two balls in this frame can be obtained using the Galilean transformation of
velocities
.
vvV
′
=−
So,
(
)
(
)
4.0 m/s 1.0 m/s
3.0 m/s
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 Fall '09
 geller

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