Dear All,
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.)
Indeed it will:
V = v * log_e (R)
10 = 3.3 * log_e (R)
R = e^(10/3.3)
R = 20.7
QED
So far, so good.
Clarke then states 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."
So, by my calculations:
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 on earth (or perhaps more appropriately, how in space) does the necessary R come to be 37 to 1?
Where am I going wrong?
Thanks in advance.
MB
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.)
Indeed it will:
V = v * log_e (R)
10 = 3.3 * log_e (R)
R = e^(10/3.3)
R = 20.7
QED
So far, so good.
Clarke then states 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."
So, by my calculations:
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 on earth (or perhaps more appropriately, how in space) does the necessary R come to be 37 to 1?
Where am I going wrong?
Thanks in advance.
MB