Bukhovtsev-et-al-Problems-in-Elementary-Physics

202 ANSWERS AND SOLUTIONS
The wagon loses its kinetic energy in view of the fact that the force of
friction f actlng on it performs negative work
MV~_Mu'_fS
2 2-
where S is the distance travelled by the wagon.
The body acquires kinetic energy because the force of friction acting on it
performs positive work
mu l
2=ls
Here s is the distance travelled by the body.
It is easy to see that the change in the kinetic energy of the system
Mv~ _
[MU 2 2]
+ mU =/ (S-s) (2)
2 2 2
is equal to the force of friction multiplied by the motion of the body relattve
to the cart.
It follows from equations (1) and (2) that
mMv~
Since S-s ~ I. then 1~ 2/ (M +m)
mMv~
S-s=2f(M +m)
Bearing in mind that f=kmg. we have l;a. 2kg7~~+m) ·
128. The combustion of the second portion increases the velocity of the
rocket v by Av. Since combustion is instantaneous, then according to the law
of conservation of momentum,
(M +m) v=M (v+t\v)+m (v-u)
where m is the mass of a fuel portion, M the mass of the rocket without
fuel, u the outflow velocity of the gases relative to the rocket.
The velocity increment of the rocket Av=Z u does not depend on the
veloci ty v before the second portion of the fuel burns. On the contrary, the
increment in the kinetic energy of the rocket (without fuel)
se, M(vtliV)1 _M;I=mu(2~u+v)
wi11 be the greater, the higher is v.
The maximum altitude of the rocket is determined by the energy it receives.
For this reason the second portion of the fuel can be burnt to the greatest
advantage when the rocket attains its maximum velocity. i. e.• directly after
the first portion is ejected. Here the greatest part of the mechanical energy
produced by the combustion of the fuel will be imparted to the rocket, while
the mechanical energy of the combustion products will be minimum.