Conservation of linear momentum provides appropriate analysis framework for analyzing motion of a rocket.

The important difference in the approach here vis-a-vis application of Newton’s second is that we can consider “rocket” and “ejected gas mass” as components of a closed, isolated system. Though, the mass of the rocket is varying with time, but the mass of the “rocket – gas mass” system is constant. This fact eliminates the complexity resulting from varying mass.

It is easy to visualize that “thrust” on the rocket should be greater than forces like gravity and air resistance, which are opposing its motion. Since rocket acquires great enough velocity and goes beyond the influence of opposing forces quickly, it is intuitive to study motion in ideal condition, when no external force other than “thrust” operates on it. This enables us to draw a base case, which can be appropriately modified by taking other external forces into account, if so required.

Thus, system is closed and isolated. The net external force is zero. As such, the linear momentum of the system is conserved.

For the analysis of the motion of a rocket, we shall make few simplifying assumptions :

- There is no external force like gravity and air resistance.
- The motion is taking place in one dimension.
- Fuel is consumed at constant rate.
- Gas is ejected at constant relative velocity with respect to rocket.

** Reference to different velocities **

Conventionally, we represent velocity of rocket with respect to ground by symbol “v”, velocity of escaping gas with respect to ground by “u” and relative velocity of rocket with respect to escaping gas by “

The absolute velocities of the rocket and ejected gas mass are shown in the figure :

Absolute velocities |
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The relative velocity of rocket with respect to escaping gas is equal to difference of absolute velocities with respect to ground. It may also be emphasized here that relative velocity of rocket with respect to escaping gas and relative velocity of escaping gas with respect to rocket are equal and opposite to each other.

** Time rate of change in mass **

The expression “

Change in mass |
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** Conservation of linear momentum **

We analyze motion of “rocket and ejected mass” system in the inertial frame of reference of ground. There is no difficulty in the interpretation as system is a single entity without any variation in mass.

At a given instant, t = t, let “m” be the mass of the rocket and “v” be the velocity of the rocket.

We, now, consider the situation at a time instant t’ = t + dt, after a small time interval, “dt”. Let “m+dm” be the mass of the rocket, v+dv be the absolute velocity of the rocket with respect to ground, “-dm” be the mass of the ejected gas and “u” be the absolute velocity of the gas mass with respect to ground.

Velocities of the components of the system |
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Now, applying conservation of linear momentum, we have :

This form of equation, however, is not very useful as it consists of absolute velocity of escaping gas, which is variable and is difficult to be measured. We need to convert this velocity in terms of relative velocity of rocket with respect to ejected mass (

The relative velocity of rocket with respect to ejected gas is equal to the magnitude of relative velocity of ejected mass with respect to rocket. This later velocity i.e. relative velocity of ejected mass with respect to rocket is actually the velocity that can be calibrated for different time rates of mass ejection at the ground.

From consideration of relative motion, we know that

In the figure, negative of the velocity of gas mass is applied to both components of the system to obtain relative velocity of the rocket with respect to ejected gas mass.

Relative velocity of the rocket |
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Rearranging, we have :

Substituting this expression for “u” in the equation of conservation of linear momentum, we have :

Dividing both sides by “dt”, we have :

We have discussed the expression on the right hand side of the equation in the module named “Force and invariant mass”. This term was found to be the “cause” element in Newton’s second law of motion known as “thrust”. The thrust, “T”, is a force that results from exchange of mass between rocket and its surrounding. It acts on the rocket in the direction opposite to the direction in which gas escapes from the rocket i.e. in the direction of motion of the rocket.

This equation is the governing expression for the motion of a rocket in the absence of other external forces.