A person moves by pushing off from the Earth; a boat sails because the rowers push against the water with their oars; a ship also pushes against the water, only not with oars but with propellers; a train moving on rails and an automobile also push off from the Earth—remember how hard it is for an automobile to get started on an icy road. Thus, pushing off from a support seems to be a necessary condition for motion; even an airplane moves by pushing the air with its propeller.
But is it really? Might there not be some intricate means of moving without pushing off from anything. If you ice-skate, you can easily convince yourself on the basis of your experience that such motion is quite possible. Pick up a heavy stick and get on the ice. Throw the stick forward—what will happen? You will glide backwards, although the thought of pushing against the ice with your foot didn’t even cross your mind.
Recoil, which we have just studied, yields us the clue to carrying out motion without support, without pushing off. Recoil presents a possibility of accelerating motion even in a vacuum, where there really is absolutely nothing to push off from.
The recoil caused by a steam jet being driven out of a vessel (the reaction of the jet) was used back in Ancient Times for creating curious toys. An ancient steam turbine invented in the second century B.C. is pictured in Figure 1. A spherical cauldron was supported by a vertical axis. Escaping from the cauldron through elbow-shaped pipes, the steam pushed these pipes in the opposite direction, and the sphere rotated.
These days the use of jet propulsion has already gone far beyond the realm of the creation of toys and the collection of interesting observations. The twentieth century is sometimes called the century of atomic energy, but with no less reason one could call it the century of jet propulsion, since the far-reaching consequences of the use of powerful jet engines can scarcely be exaggerated. This is not only a revolution in aircraft construction but the beginning of mankind’s contact with the Universe.
The principle of jet propulsion permits the creation of airplanes moving with a speed of several thousand kilometres per hour, flying missiles rising hundreds of kilometres above the Earth, artificial Earth satellites and cosmic rockets carrying out interplanetary flights.
A jet engine is a machine from which gases formed by the combustion of fuel are ejected with great force. The rocket moves in the direction opposite to that of the gas stream.
How strong is the thrust carrying the rocket off into space? We know that the force is equal to the change in momentum during a unit of time. According to our conservation law, the momentum of the rocket changes by the total momentum \(mv\) of the ejected gas.
This law of nature allows us to compute, for example, the relation between the force of the jet propulsion and the expenditure of fuel necessary for this. In doing so, one must assume a value for the speed of discharge of the combustion products.
Let us take, say, the following values: the gases are ejected with a speed of \(2000\,\mathrm{m/s}\) at the rate of 10 tons per second. Then the force in the jet propulsion will be about \(2\times 10^{12}\) dyn, i.e. approximately \(2000\) tonf.
Let us determine the change in speed of a rocket moving in interplanetary space.
The momentum of the mass \(\Delta M\) of gas ejected with speed \(u\) is equal to \(u \Delta M\). The momentum of a rocket of mass \(M\) will increase by the amount \(M \Delta V\). According to our conservation law, these two quantities must be equal to each other: \[u \Delta M = M \Delta V, \,\, \textrm{i.e.} \,\, \Delta V = u \frac{\Delta M}{M}\] However, if we wish to compute the speed of a rocket when the ejected mass is comparable to the mass of the rocket, the formula we have derived turns out to be useless. In fact, it assumes that the mass of the rocket is constant. However, the following important result remains valid: identical relative changes in mass lead to one and the same change in speed.
A reader acquainted with the basics of integral calculus will at once obtain the true formula. It has the form \[V = u \ln \frac{M_\text{in}}{M} = 2.3 \, u \ln \frac{M_\text{in}}{M} \label{rocket-eq}\] If you use a slide rule, you will find that when the mass of the rocket is cut in half, its speed will reach \(0.7u\).
In order to raise the speed of the rocket to \(3u\), it is necessary to burn up a mass \(m = (19/20) M\). This means that only one-twentieth of the mass of the rocket can be preserved if we wish to raise its speed to \(3u\), i.e. to 6–8 kW/s.
In order to attain a speed of \(7u\), the mass of the rocket must decrease by 1000 times during the speed-up.
These calculations warn us against striving to increase the mass of the fuel which can be put in the rocket. The more fuel we take, the more we must burn. For a given speed of gas ejection, it is very difficult to achieve an increase in the speed of the rocket.
The increase in the speed of gas ejection is the basic means of attaining high rocket speeds. In this respect, a significant role must be played by the application to rockets of engines running on a new atomic fuel.
For a constant speed of gas ejection, a gain in speed with the same mass of fuel is obtained by using multistage rockets. In a single-stage rocket, the mass of the fuel decreases, but the empty tanks keep moving with the rocket. An additional energy is required to accelerate the mass of the unnecessary fuel tanks. It would be expedient to throwaway the fuel tanks whose fuel has been consumed. In modern multi-stage rockets, not only are the fuel tanks and piping thrown away but also the engines of the used stages.
Of course, it would be best to continuously throwaway the unnecessary mass of the rocket. Such a construction does not yet exist. The take-off weight of a three-stage rocket can be made six times less than that of a single- stage rocket with the same “ceiling” A “continuous” rocket would be more profitable in this sense than a three-stage rocket by an additional 15%.