Rocket Equation

Here, M0 is the mass of the rocket at ignition, M is the current mass of the rocket and ve is the effective exhaust velocity. This simple formula is the basis for all rocket propulsion. In a rocket, its mass keeps on decreasing as it moves forward.

The same force which projects the exhaust gases outward is the same force that propels the rocket forward according to Newton’s third law. This accelerating force is given by F=mve, where m is the propellant mass flow rate and ve is the effective exhaust velocity. Newton’s second law can be applied to dynamical systems, so it can be applied to systems with decreasing mass like a rocket.

How was Tsiolkovsky’s Rocket Equation derived?

The resultant formula which Tsiolkovsky obtained for the rocket with velocity V is known as Tsiolkovsky’s rocket equation. The velocity increases with time as the propellant is burned. It also depends on the natural logarithm of the initial mass to present mass, i.e., on how much the propellant has been burned. For a fixed amount of propellant burnt, it depends on how fast the propellant has been expelled i.e., the effective exhaust velocity.

In most cases, the final velocity of the rocket is to be known, hence the appropriate value of mass ratio to be used is when all the propellant has been burnt. Hence the final rocket velocity depends on only two parameters; the mass ratio and effective exhaust velocity. Surprisingly, it does not depend on the thrust, the time of propellant burn or the size of the rocket engine, or any other parameter.

Why in Rockets BIGGER is BETTER?

Clearly, a higher exhaust velocity means a higher rocket velocity, hence much of the effort in rocket engine design goes to increasing the exhaust velocity. It happens that exhaust velocity, within a narrow range of variability related to engine design, depends just on the chemical nature of the propellant.

Gunpowder, or the other propellants used in rockets until the 19th Century provided an exhaust velocity only up to 2000m/s. The most advanced chemical rockets today give an exhaust velocity of at best, 4500m/s. There is no scope for improvement because this is close to the theoretical limit for energy extraction.

For a fixed propellant combination, the only way to increase velocities is to increase mass ratios. The mass ratio is defined as the ratio of the total mass of the rocket fully loaded with fuel to the rocket with its fuel tanks empty. A mass ratio of 5 means that 80% of the initial rocket mass is fuel.

Compare this to a typical car; taking the car mass to be 1000 kg and the fuel tank to hold 40 kg of fuel, the mass ratio is 1.004. So rocket’s mass ratio is extremely high when compared to any other vehicle. Another comparison with a car can be made regarding the payload.

A typical car caries 5 passengers, assuming each passenger to be 60 kg, so the total payload for the car is 240 kgs with its mass being 1000 kg. Contrast this with a rocket weighing thousands of tonnes and carrying payloads of some thousands. The mighty Saturn V rocket which put the men on the moon weighed 3,000 tonnes most of which was fuel.

What can we infer about the Rocket from this Graph?

We can see from the graph that a rocket can travel faster than the speed of its exhaust. This may sound counterintuitive if we think a rocket goes forward with its exhaust pushing against something. The exhaust is not pushing against anything at all, after the exhaust has left the rocket, it has further no effects on the rocket.

All the action takes place inside the rocket engine, with the exhaust gasses pushing against the rocket nozzle. Even though the exhaust velocities are restricted to theoretical max values, the actual speed of the rocket can be still greater, thanks to its mass ratio. So, a person standing on the ground can actually see both the rocket and its exhaust moving in the same direction overhead.

Can Rockets work in a vacuum?

Another point to be noted is that the accelerating force is independent of the speed of the rocket. So, with a very large mass ratio, very high speeds can be obtained. Rocket can also work in a vacuum since it caries its oxidizer along with the fuel, it doesn’t need an air intake like for a jet engine.

The rocket actually works better in a vacuum because the air pressure retards the exhaust and hence reduced the thrust. Tsiolkovsky also calculated the velocity needed in order to reach space and found that the rocket equation had a theoretical limit. It can be seen from the graph that after a certain point, increasing the mass ratio has little effect on the rocket velocity.

What did Tsiolkovsky conclude from this Rocket Equation regarding Space Travel?

From the curve, if we consider the rocket with an exhaust velocity of 1km/s, then the rocket velocity which can be reasonably achieved is 3km/s. A higher mass ratio would give higher rocket velocity but only a little. A mass ratio of 10 is almost impossible to achieve, that too with a sophisticated high exhaust rocket engine.

Tsiolkovsky knew that 11km/s was needed to leave Earth, but the gunpowder could only provide an exhaust velocity of about 2km/s, at this rate, it was almost impossible to leave Earth. So, the only thing Tsiolkovsky could do was increase exhaust velocity, this could be done by changing the combustion temperature and molecular weight of the propellants.

He found out that liquid fuel rockets powered by hydrogen and oxygen could provide exhaust velocities in excess of 4km/s and this discovery made space travel a real possibility than an imaginary fantasy.

Reference

Turner, M. J. L. (2005). Rocket and Spacecraft Propulsion: Principles, Practice and New Developments (Second Edition). Chichester, UK: Springer-Praxis.

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