Numbers in the appendix of Arthur C. Clarke's 1945 classic paper

[MB2012]

Homework Statement

This question relates to the numbers in the appendix of Arthur C. Clarke's 1945 classic paper (Wireless World, Oct. 1945, pp. 305-308).

The appendix (on page 308 of the paper) outlines some basic principles of rocket design, including Tsiolkovsky's fundamental equation of rocket motion which is given as:

V = v log_e (R)

where V is the final velocity of the rocket, v is the exhaust velocity, and R is the ratio of initial mass to final mass (payload plus structure).

Clarke states that: "If we assume v to be 3.3 km/sec. R will be 20 to 1." (For V = 10 km/sec.)

Clarke goes on to state that due to the rocket's finite acceleration, it loses velocity as a result of gravitational retardation and the necessary ratio R_g is increased to

R_g = R * ( (a + g) / (a) )

where a is the rocket's acceleration and g is the acceleration due to gravity.

Clarke goes on to say that (and this is where my problem is): "For an automatically controlled rocket a would be about 5g and so the necessary R would be 37 to 1."

How does the necessary R come to be 37 to 1?

Homework Equations

Tsiolkovsky's fundamental equation of rocket motion:

V = v log_e (R)

where V is the final velocity of the rocket, v is the exhaust velocity, and R is the ratio of initial mass to final mass (payload plus structure).

R_g = R * ( (a + g) / (a) )

where a is the rocket's acceleration and g is the acceleration due to gravity.

The Attempt at a Solution

V = v * log_e (R)

10 = 3.3 * log_e (R)

R = e^(10/3.3)

R = 20.7

QED (R is approximately 20 to 1)

R_g = R * ( (a + g) / (a) )

R_g = R * ( (5g + g) / (5g) )

R_g = R * (6g / 5g)

R_g = 1.2 * R

(EQUIVALENTLY: R = 0.83 * R_g)

How does R become 37 to 1?

 

[anathema]

To determine how the necessary R comes to be 37 to 1, we need to first understand the concept of specific impulse. This is a measure of the efficiency of a rocket engine and is defined as the thrust produced by the engine divided by the rate of propellant consumption. In other words, it is a measure of how much thrust is produced per unit of propellant. In this case, the specific impulse is given by the exhaust velocity (v).

Using Tsiolkovsky's equation, we can see that for a given final velocity (V), the necessary R (ratio of initial mass to final mass) will increase as the exhaust velocity decreases. This is because a lower exhaust velocity means that more propellant is needed to achieve the same final velocity.

Now, in the case of an automatically controlled rocket, the acceleration (a) will be much higher than the acceleration due to gravity (g). This is because the rocket is designed to accelerate quickly and efficiently, so a higher acceleration is needed.

Using the equation for R_g, we can see that as the acceleration (a) increases, the necessary R also increases. This is because a higher acceleration means that the rocket will need more propellant to overcome the gravitational pull and maintain its velocity.

So, in the case of an automatically controlled rocket with an acceleration of 5g, the necessary R will be significantly higher than 20 to 1. In fact, using the equation R = 0.83 * R_g, we can see that R will be approximately 37 to 1. This means that for every 1 unit of initial mass, the rocket will need 37 units of final mass (payload plus structure) to achieve a final velocity of 10 km/sec.

In conclusion, the necessary R of 37 to 1 comes from the combination of a high acceleration (5g) and a relatively low specific impulse (3.3 km/sec), both of which are necessary for an automatically controlled rocket to reach a final velocity of 10 km/sec.