Answer to Question #156648 in Classical Mechanics for Afroja Akhi

Question

A spaceship of mass m has velocity v in the positive x direction of an inertial reference frame. A mass dm is fired out the rear of the ship with constant exhaust velocity (-v0) with respect to the spaceship. a) using conservation of momentum, show that dv/v0 = dm/m, b) By integration, find the dependence of v on m if v1 and m1 are the initial values. c) Can the acceleration be constant if dm/dt, the burning rate is constant.

Answer 1

a)

$$p_i=p_f\\mv=(m-dm_g)(v_dv)+dm_g(v-u)\\mdv=dm_gdv+dm_gv$$

Now, $dm_g$ and dv are each very small; thus, their product is very, very small, much smaller than the other two terms in this expression. We neglect this term, therefore, and obtain:

$$mdv=dm_gv$$

Our next step is to remember that, since $dm_g$ represents an increase in the mass of ejected gases, it must also represent a decrease of mass of the rocket:

$$dm_g=-dm\\mdv=-dmu$$

b)

$$v=v_1+u\ln{\frac{m_1}{m}}$$

c) Yes.