3.30.
CHAPTER 3, PROBLEM 30
247
3.30
Chapter 3, Problem 30
Problem:
A particle of mass
m
is moving with constant velocity
v
=
V
i
when it suddenly explodes.
After the explosion, there are two fragments.
One fragment moves at an angle
2
θ
to the horizontal
as shown with speed
3
2
V
. Its mass is
λ
M
, where
λ
<
1
is a constant. The other fragment of mass
(1
−
λ
)
M
, moves at an angle
θ
to the horizontal as shown, also with speed
3
2
V
. Solve for
λ
and
θ
.
M
v
=
V
i
λ
M
(1
−
λ
)
M
2
θ
θ
3
2
V
3
2
V
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Solution:
The explosion gives rise to internal forces that cause the particle fragments to move away from
the original path along which the particle moved. Since there are no external applied forces, momentum is
conserved, wherefore
MV
i
=
3
2
λ
MV
(cos 2
θ
i
+ sin 2
θ
j
) +
3
2
(
λ
−
1)
MV
(cos
θ
i
−
sin
θ
j
)
So, in terms of
x
and
y
components, we have
MV
=
3
2
λ
MV
cos 2
θ
+
3
2
(1
−
λ
)
MV
cos
θ
0
=
3
2
λ
MV
sin 2
θ
−
3
2
(1
−
λ
)
MV
sin
θ
Using the fact that
sin 2
θ
= 2 sin
θ
cos
θ
, the second equation becomes
0 = 3
λ
MV
sin
θ
cos
θ
−
3
2
(1
−
λ
)
MV
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 Spring '06
 Shiflett
 Force, Mass, Momentum, ... ..., λ. 8λ −, M V

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