CLASSICAL MECHANICS
Specificity of Newton's laws for objects with variable mass

Rocket motion

Up to this point we have viewed only problems in which the mass of the system remains constant. Newton himself recognized that situations in which the mass is not constant were still covered by his formulation of the Second Law of Motion even though such situations are less common. These cases always demand that we know how to handle differential equations.

One example where the mass does change is the case of rockets. Rockets are perfect for space travel because they carry their fuel and oxygen supply with them. In fact, most of the mass of the rocket on the ground is in the form of fuel and oxidizer. In space, the burning fuel is ejected from the rear of the rocket. This action produces a reaction force on the rocket body which propels it forward. Note that there is no need for any air to "push against" for the rocket to work. Newton's Third Law assures us that ejection of an object from the system MUST propel the system in the opposite direction. The propulsive force is referred to as the thrust of the rocket.

To derive the equation which describes rocket motion, we begin by translating words about the rocket's actions into equations. We begin by noting that the rocket expels gases from burning fuel. If we imagine that the rocket is in outer space, far from the earth, then there are no drag forces or gravity so there are no external forces on the rocket or the gases it expels. Therefore, if we consider the rocket and the exhaust to be a system, then, the momentum of this system is constant with time. Consider the figure below:

ΔP_rocket=Δ(Mv) = (ΔM)v + M(Δv)

This equation says that the rocket changes momentum due to its loss in mass (which is just the mass of gas expelled) times its initial velocity plus the mass of the rocket times its change in velocity. To get the momentum of the expelled gas, we need to know its velocity. Typically, rocket engineers measure the speed relative to the rocket body at which gases are ejected as a way of characterizing the engine. If we call the speed relative to the rocket, u_ex, then the speed of exhaust relative to us as observers is v-u_ex. The total mass of the exhaust for this time is just , the mass lost by the rocket body, hence the net change in momentum of the exhaust is

ΔP_ex = -ΔM(v-u_ex)

where the minus sign indicates that the momentum change in the exhaust is opposite to that of the rocket body motion. Since momentum is conserved, the total changes in momentum of the rocket body and exhaust gases must cancel

We can develop this last equation into a differential equation by noting that our expression becomes more and more accurate a description of the rocket's motion for any given time if we let Δt -> 0. Therefore, the rocket equation that describes the motion for any time t is simply

where F_ext is the sum of the external forces. The force, M(dv/dt) is what we call the thrust of the rocket.

Solving of the equations gives us solution for the rocket velocity:

If M₀ is the initial mass of the rocket and M_fuel is the mass of the fuel at liftoff, then M₀ - M_fuel represents the mass of the payload. The final burnout velocity of the rocket is then given by

This equation ignores air resistance, the weight of the rocket body itself, and does not take into account the fact that the force of gravity reduces with height (we need integration to do that problem), but nevertheless is pretty accurate for characterizing most rocket launches, including the Space Shuttle.