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326 CHAPTER 10 Systems of Particles

COMMENT: The distances moved by the astronaut and by the spacecraft are in
the inverse ratio of their masses.The astronaut (of small mass) moves a large dis-
tance, and the spacecraft (of large mass) moves a smaller distance.This is the result
of the accelerations that the pull of the cord gives to these bodies: with forces of equal
magnitudes, the accelerations of the astronaut and spacecraft are in the inverse
ratio of their masses. However, our method of calculation based on the fixed posi-
tion of the center of mass gives us the final positions directly, without any need to
examine accelerations.

A projectile is launched at some angle with respect to the
EXAMPLE 10
horizontal, 0 90 . Just as it reaches its peak,it explodes
into two pieces.The explosion causes a first,rear piece to come to a momentary stop,
and it simply drops, striking the ground directly below the peak position.The explo-
sion also causes the speed of the second piece to increase, and it hits the ground a
distance five times further from the launch point than the first piece (see Fig. 10.24).
If the original projectile had a mass of 12.0 kg, what are the masses of the pieces?

SOLUTION: Because the explosion does not produce external forces, the center
of mass continues on its original path, a parabolic trajectory which strikes the
ground at the range x max , given by Eq. (4.43).The peak of the parabolic trajectory
occurs at half this distance; thus the first piece, of some mass m , hits the ground
1
1
a distance x max from the launch point. We are also told that the second piece, of
2
1
2
mass m , hits the ground a distance 5 x from the launch point. The two
2 max
pieces will reach the ground at the same instant, since this explosion affected only
each piece’s horizontal momentum. If we take our origin at the launch point, the
x component of the center of mass is thus
m x m x m x >2 5m x >2
2 max
1 1
2 2
x x 1 max
CM max
m m m m
1 2 1 2
We can divide both sides of this equation by x max and rearrange to obtain
m 3m
1 2
Since we know the total mass is m m 12.0 kg, or 4m 12.0 kg, we obtain
1
2
2
m 9.0 kg and m 3.0 kg
1 2

Fragments are at the
same height.

x max
original projectile range
FIGURE 10.24 A projectile explodes at its apex. The rear fragment simply drops, and the forward
piece lands five times further from the launch point.