33. As hinted in the problem statement, the velocity v of the system as a whole – when the spring reaches
the maximum compression x
m
– satisfies m
1
v
1i
+ m
2
v
2i
= (m
1
+ m
2
)v. The change in kinetic energy of
the system is therefore
∆K
=
1
2
(m
1
+ m
2
)v
2
−
1
2
m
1
v
2
1i
−
1
2
m
2
v
2
2i
=
(m
1
v
1i
+ m
2
v
2i
)
2
2 (m
1
+ m
2
)
−
1
2
m
1
v
2
1i
−
1
2
m
2
v
2
2i
which yields ∆K =
−35 J. (Although it is not necessary to do so, still it is worth noting that algebraic
manipulation of the above expression leads to
|∆K| =
1
2
m
1
m
2
m
1
+m
2
v
2
rel
where v
rel
= v
1
−v
2
). Conservation
of energy then requires
1
2
kx
2
m
=
−∆K =⇒ x
m
=
−2∆K
k
=
−2(−35)
1120
which gives the result x
m
= 0.25 m.