A Classical Approach of Newtonian Mechanics

6 CONSERVATION OF MOMENTUM 6.4 Rocket science

rocket

propellant

Figure 50: A rocket.

t t+dt

v (^) M+m M+m+dm v+dv

• dm v-u^

Figure 51: Derivation of the rocket equation.

Let us attempt to find the equation of motion of a rocket. Let M be the fixed

mass of the rocket engine and the payload, and m(t) the total mass of the pro-

pellant contained in the rocket’s fuel tanks at time t. Suppose that the rocket

engine ejects the propellant at some fixed velocity u relative to the rocket. Let

us examine the rocket at two closely spaced instances in time. Suppose that at

time t the rocket and propellant, whose total mass is M + m, are traveling with

instantaneous velocity v. Suppose, further, that between times t and t + dt the

rocket ejects a quantity of propellant of mass −dm (n.b., dm is understood to
be negative, so this represents a positive mass) which travels with velocity v − u

(i.e., velocity −u in the instantaneous rest frame of the rocket). As a result of the

fuel ejection, the velocity of the rocket at time t + dt is boosted to v + dv, and its

total mass becomes M + m + dm. See Fig. 51.

Now, there is zero external force acting on the system, since the rocket is