# The Travel Velocity and the Efficiency of Rocket Vehicles

It is very important and characteristic of the reaction vehicle that the travel velocity may not be selected arbitrarily, but is already specified in general due to the special type of its propulsion. Since continual motion of a vehicle of this nature occurs as a result of the fact that it expels parts of its own mass, this phenomenon must be regulated in such a manner that all masses have, if possible, released their total energy to the vehicle following a successful expulsion, because the portion of energy the masses retain is irrevocably lost. As is well known, energy of this type constitutes the kinetic force inherent in every object in motion. If now no more energy is supposed to be available in the expelling masses, then they must be at rest visavis the environment (better stated: visavis their state of motion before starting) following expulsion. In order, however, to achieve this, the travel velocity must be of the same magnitude as the velocity of expulsion, because the velocity, which the masses have before their expulsion (that is, still as parts of the vehicle), is just offset by the velocity that was imparted to them in an opposite direction during the expulsion (Figure 16). As a result of the expulsion, the masses subsequently arrive in a relative state of rest and drop vertically to the ground as free falling objects.

Figure 16. The travel velocity is equal to the velocity of expulsion. Consequently, the velocity of the expelled masses equals zero after the expulsion, as can be seen from the figure by the fact that they drop vertically.

Key: 1. Expelled masses; 2. Velocity of expulsion; 3. Travel velocity; 4. Cart with reactive propulsion

Under this assumption in the reaction process, no energy is lost; reaction itself works with a (mechanical) efficiency of 100 percent (Figure 16). If the travel velocity was, on the other hand, smaller or larger than the velocity of expulsion, then this "efficiency of reactive propulsion" would also be correspondingly low (Figure 17). It is completely zero as soon as the vehicle comes to rest during an operating propulsion.

This can be mathematically verified in a simple manner, something we want to do here by taking into consideration the critical importance of the question of efficiency for the rocket vehicle. If the general expression for efficiency is employed in the present case: "Ratio of the

Figure 17. The travel velocity is smaller (top diagram) or larger (lower diagram) than the velocity of expulsion. The expelled masses still have, therefore, a portion of their velocity of expulsion (top diagram) or their travel velocity (lower diagram) following expulsion, with the masses sloping as they fall to the ground, as can be seen in the figure.

Key: 1. Expelled masses; 2. Velocity of expulsion; 3. Travel velocity; 4. Cart with reactive propulsion energy gained to the energy expended", then the following formula is arrived at as an expression for the efficiency of the reaction hr as a function of the instantaneous ratio between travel velocity v and the velocity of repulsion c.

In Table 1, the efficiency of the reaction hr is computed for various values of this v/c ratio using the above formula. If, for example, the v/c ratio was equal to 0.1 (i.e., v=0.1 c, thus the travel velocity is only onetenth as large as the velocity of expulsion), then the

efficiency of the reaction would only be 19 percent. For v/c=0.5 (when the travel velocity is onehalf as large as the velocity of repulsion), the efficiency would be 75

Table 1

Ratio of the travel Efficiency of the

velocity v to the Reaction hr

velocity of expulsion c

v/c hr in percentages

(roundedup)

0 0 0

0.01 0.0199 2

0.05 0.0975 10

0.1 0.19 19

0.2 0.36 36

0.5 0.75 75

0.8 0.96 96

1 1 100

1.2 0.96 96

1.5 0.75 75

1.8 0.36 36

2 0 0

2.5 1.25 125

3 3 300

4 8 800

5 15 1500

percent, and for v/c=1 (the travel velocity equals the velocity of expulsion)in agreement with our previous considerationthe efficiency would even be 100 percent. If the v/c ratio becomes greater than 1 (the travel velocity exceeds the velocity of expulsion), the efficiency of the reaction is diminished again and, finally, for v/c=2 it again goes through zero and even becomes negative (at travel velocities more than twice as large as the velocity of expulsion).

The latter appears paradoxical at first glance because the vehicle gains a travel velocity as a result of expulsion and apparently gains a kinetic force as a result! Since the loss of energy, resulting through the separation of the expulsion mass loaded very heavily with a kinetic force due to the large travel velocity, now exceeds the energy gain realized by the expulsion, an energy loss nevertheless results for the vehicle from the entire processdespite the velocity increase of the vehicle caused as a result. The energy loss is expressed mathematically by the negative sign of the efficiency. Nonetheless, these efficiencies resulting for large values of the v/c ratio have, in reality, only a more or less theoretical value.

It can, however, clearly and distinctly be seen from the table how advantageous and, therefore, important it is that the travel velocity approaches as much as possible that of the expulsion in order to achieve a good efficiency of reaction, but slight differences (even up to v=0.5 c and/or v=1.5 c) are, nevertheless, not too important because fluctuations of the efficiency near its maximum are fairly slight. Accordingly, it can be stated that the optimum travel velocity of a rocket vehicle is approximately between onehalf and one and onehalf times its velocity of expulsion.

When, as is the case here, the reaction vehicle is a rocket vehicle and consequently the expulsion of masses takes place through appropriate combustion and exhausting of propellants carried on the vehicle, then, in the sense of the requirement just identified, the travel velocity must be as much as possible of the same magnitude as the exhaust velocity (Figure 18). To a certain extent, this again requires, however, that the travel velocity conforms to the

Figure 18. For a rocket vehicle, the travel velocity must as much as possible be equal to the exhaust velocity.

Key: 1. Exhausted gases of combustion; 2. Exhaust velocity; 3. Travel velocity; 4. Cart with rocket propulsion

type of propellants used in each case, because each has its own maximum achievable exhaust velocity.

This fundamental requirement of rocket technology is above all now critical for the application of rocket vehicles. According to what has already been stated, the velocity of repulsion should then be as large as possible.

Actually, the possible exhaust velocities are thousands of meters per second and, therefore, the travel velocity must likewise attain a correspondingly enormous high value that is not possible for all vehicles known to date, if the efficiency is supposed to have a level still useable in a practical application.

This can be clearly seen from Table 2, in which the efficiencies corresponding to the travel velocities at various velocities of expulsion are determined for single important travel velocities (listed in Column 1). It can be seen from Column 2 of the table, which lists the efficiency of reaction, how uneconomical the rocket propulsion is at velocities (of at most several hundred kilometers per hour) attainable by our present vehicles.

This stands out much more drastically if, as expressed in Column 3, the total efficiency is considered. This is arrived at by taking into account the losses that are related to the generation of the velocity of expulsion (as a result of combustion and exhausting of the propellants). These losses have the effect that only an exhaust velocity smaller than the velocity that would be theoretically attainable in the best case for those propellants can ever be realized in practice. As will subsequently be discussed in detail, the practical utilization of the propellants could probably be brought up to approximately 60 percent. For benzene by way of example, an exhaust velocity of 3,500 meters per second at 62 percent and one of 2,000 meters per second at 20 percent would result. Column 3 of Table 2 shows the total efficiency for both cases (the efficiency is now only 62 percent and/or 20 percent of the

Table 2

1 2 3

Travel Efficiency of the Reaction Total

velocity Efficiency

of the

v vehicle

propulsion

h=hrhi

for benzene

and liquid

in oxygen as

propellants

Expressed in percentages for the following

velocities of repulsion c in m/sec:

km/h m/s 1000 2000 2500 3000 3500 4000 5000 2000 3500

40 11 2.2 1.2 0.9 0.7 0.6 0.5 0.4 0.2 0.4

100 28 4.6 2.8 2.2 1.8 1.6 1.4 1.2 0.6 1

200 56 11 5.5 4.5 3.8 3.2 2.8 2.2 1.1 2

300 83 16 8 6.5 5.5 4.7 4 3.4 1.6 3

500 140 26 13 11 9 8 7 5.5 2.7 5

700 200 36 19 15 13 11 10 8 4 7

1000 300 51 28 23 19 16 14 12 6 10

1800 500 75 44 36 31 27 23 19 9 17

3000 1000 100 75 64 56 50 44 36 15 31

5400 1500 75 94 84 75 67 60 51 19 42

7200 2000 0 100 96 89 81 75 64 20 50

9000 2500 125 94 100 97 92 86 75 19 57

10800 3000 300 75 96 100 98 94 84 15 61

12600 3500 525 44 84 97 100 99 91 9 62

14400 4000 800 0 64 89 98 100 96 0 61

18000 5000 1500 125 0 56 81 94 100 25 50

21600 6000 300 96 0 50 75 96 61 31

25200 7000 520 220 77 0 44 70 111 0

28800 8000 800 380 175 64 0 64 160 40

36000 10000 1500 800 440 250 125 0 300 160

45000 12500 1500 900 560 350 125 350

corresponding values in Column 2, in the sense of the statements made).

As can be seen from these values, the total efficiencyeven for travel velocities of many hundreds of kilometers per houris still so low that, ignoring certain special purposes for which the question of economy is not important, a farreaching practical application of rocket propulsion can hardly be considered for any of our customary means of ground transportation.

On the other hand, the situation becomes entirely different if very high travel velocities are taken into consideration. Even at supersonic speeds that are not excessively large, the efficiency is considerably better and attains even extremely favorable values at still higher, almost cosmic travel velocities in the range of thousands of meters per second (up to tens of thousands of kilometers per hour), as can be seen in Table 2.

It can, therefore, be interpreted as a particularly advantageous encounter of conditions that these high travel velocities are not only possible (no resistance to motion in empty space!) for space vehicles for which the reaction represents the only practical type of propulsion, but even represent an absolute necessity. How otherwise could those enormous distances of outer space be covered in acceptable human travel times? A danger, however, that excessively high velocities could perhaps cause harm does not exist, because we are not directly aware whatsoever of velocity per se, regardless of how high it may be. After all as "passengers of our Earth," we are continually racing through space in unswerving paths around the sun at a velocity of 30,000 meters per second, without experiencing the slightest effect. However, the "accelerations" resulting from forced velocity changes are a different matter altogether, as we will see later.

Table 3 permits a comparison to be made more easily

Table 3

Kilometers Meters Kilometers

per hour per second per second

km/hour m/sec km/sec

5 1.39 0.00139

10 2.78 0.00278

30 8.34 0.00834

50 13.9 0.0139

70 19.5 0.0195

90 25.0 0.0250

100 27.8 0.0278

150 41.7 0.0417

200 55.6 0.0556

300 83.4 0.0834

360 100 0.1

500 139 0.139

700 195 0.195

720 200 0.2

1000 278 0.278

1080 300 0.3

1190 330 0.33

1800 500 0.5

2000 556 0.556

2520 700 0.7

3000 834 0.834

3600 1000 1

5400 1500 1.5

7200 2000 2

9000 2500 2.5

10800 3000 3

12600 3500 3.5

14400 4000 4

18000 5000 5

21600 6000 6

25200 7000 7

28800 8000 8

36000 10000 10

40300 11180 11.18

45000 12500 12.5

54000 15000 15

72000 20000 20

among the various travel velocities under consideration here something that is otherwise fairly difficult due to the difference of the customary systems of notation (kilometers per hour for present day vehicles, meters or kilometers per second for space travel).