38. Our notation (and, implicitly, our choice of coordinate system) is as follows: the mass of the original
body is m; its initial velocity is
v
0
= vˆi; the mass of the less massive piece is m
1
; ; its velocity is
v
1
= 0;
and, the mass of the more massive piece is m
2
. We note that the conditions m
2
= 3m
1
(specified in the
problem) and m
1
+ m
2
= m generally assumed in classical physics (before Einstein) lead us to conclude
m
1
=
1
4
m
and
m
2
=
3
4
m .
Conservation of linear momentum requires
m
v
0
=
m
1
v
1
+ m
2
v
2
mvˆi =
0 +
3
4
m
v
2
which leads to
v
2
=
4
3
v ˆi .
The increase in the system’s kinetic energy is therefore
∆K
=
1
2
m
1
v
2
1
+
1
2
m
2
v
2
2
−
1
2
mv
2
0
=
0 +
1
2
3
4
m
4
3
v
2
−
1
2
mv
2
=
1
6
m v
2
.