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.