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At the same moment that the total instantaneous rocket mass is m (i.e., m is the mass of the rocket body plus the mass of the fuel at that point in time), we define the rocket’s instantaneous velocity to be (in the + x -direction); this velocity is measured relative to an inertial reference system (the Earth, for example). Thus, the initial momentum of the system is
The rocket’s engines are burning fuel at a constant rate and ejecting the exhaust gases in the − x -direction. During an infinitesimal time interval dt , the engines eject a (positive) infinitesimal mass of gas at velocity ; note that although the rocket velocity is measured with respect to Earth, the exhaust gas velocity is measured with respect to the (moving) rocket. Measured with respect to the Earth, therefore, the exhaust gas has velocity .
As a consequence of the ejection of the fuel gas, the rocket’s mass decreases by , and its velocity increases by . Therefore, including both the change for the rocket and the change for the exhaust gas, the final momentum of the system is
Since all vectors are in the x -direction, we drop the vector notation. Applying conservation of momentum, we obtain
Now, 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:
Our next step is to remember that, since represents an increase in the mass of ejected gases, it must also represent a decrease of mass of the rocket:
Replacing this, we have
or
Integrating from the initial mass m i to the final mass m of the rocket gives us the result we are after:
and thus our final answer is
This result is called the rocket equation . It was originally derived by the Soviet physicist Konstantin Tsiolkovsky in 1897. It gives us the change of velocity that the rocket obtains from burning a mass of fuel that decreases the total rocket mass from down to m . As expected, the relationship between and the change of mass of the rocket is nonlinear.
In rocket problems, the most common questions are finding the change of velocity due to burning some amount of fuel for some amount of time; or to determine the acceleration that results from burning fuel.
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