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Two Stage Rocket

We know that,

That is, the velocity attained by the rocket: (i) is directly proportional to exhaust velocity, and (ii) varies logarithmically with mass ratio. Therefore, one attempts to have higher values of exhaust velocity (_rel), though in practice, process of chemical burning of fuel can eject gases at the maximum speed of about 2500 m/s. Similarly, the ratio of fuel mass to payload and casing (payload refers to final mass to be projected into space like a satellite, and casing implies the mass which contains fuel i.e. rocket engine etc.) cannot be increased too much. Hence in order to attain high velocity, we use the concept of multi-stage rocket system, which avoids the problem of higher mass ratio.

Let us consider a typical model of two stage rocket system. Suppose the mass of payload is m, that of payload + 2^nd rocket is nm, and that of payload + both (2^nd and 1^st) rockets is Nm. That is, the mass of 1^st rocket (which burns first) is (N – n)m, and that of 2^nd rocket is (n – 1) m.

Further, each rocket has a casing and fuel. Let us assume that the ratio of casing to total mass of each rocket is r. that is, mass of casing of first rocket is r (N – n) m, and that of second rocket is r (n – 1) m. In rocket propulsion, after the entire fuel of a rocket is burnt, its casing is dropped into space and then the next rocket operates.

Let us determine the final velocity of system after both the engines are fired.

Let u be the exhaust velocity. Neglecting gravity and taking v₀ = 0, the velocity v₁ of the rocket system at the end of first stage is given by

Hence, final velocity of the system is

Given particular values of m, N, and r, i.e. mass of payload, total mass of the system, and ratio of casing to rocket mass how should we change n to achieve maximum V? To find that, we put

Solving above, we get n = √N

Substituting above value of n, we find

Hence, V_max ≃ u In (√N)2 = u In N

In contrast, if we had only single stage rocket with same mass parameters, the final speed would have been

Of course, in case mass of the casing is neglected, the single stage rocket leads to same result as two stage system:

V’ = V_max = u In N.

Remember that above analysis is true when gravity is neglected and no friction from air is considered. Thus, it is fairly good approximation in outer space.