 |
where: m0 = rocket's initial mass m1 = rocket's final mass  v = change in rocket velocityve = exhaust velocity of the rocket ln = natural logarithm |
The fundamental equation of rocket propulsion is:
|
where: m0 = rocket's initial mass m1 = rocket's final mass v = change in rocket velocityve = exhaust velocity of the rocket ln = natural logarithm |
This equation lets us calculate the mass of propellant that will be required to accelerate a rocket of a given mass to a given velocity. The highest practical rocket ejection velocities are achieved by burning hydrogen with oxygen which can produce (averaged over the trajectory), an exhaust velocity ve in the order of 4000 m/s.
| You can use this calculator to find the starting mass of a rocket that will be required to accelerate a given final mass to a given change in velocity. Enter the appropriate values, and then press the calculate button. |
To achieve a low Earth orbit (200 km altitude) requires a velocity of about 7800 m/s.
However, to get to orbit from launch on the surface, you have to travel through the
Earth's atmosphere. There is a lot of atmospheric drag, as well as the force of
gravity to work against, which makes you lose roughly 1500 m/s.
Therefore the change in velocity required to go to low Earth orbit is about 9300 m/s
(as shown in the following calculation, the v = Final Velocity + Drag - Initial Velocity).
| Single Stage | Final velocity | 7800 m/s | ||
| + 1500 m/s drag forces | 1500 m/s | |||
| Initial velocity | 0 m/s | |||
| v |
9300 m/s |
Assuming this v and using the
calculator, work out the initial mass (m0) required to put 100 tonnes (the
approximate final mass (m1) of the space shuttle) into low Earth orbit if you
used a single stage rocket.
There is a serious problem here. Remember that the rocket is not entirely payload and propellant. The rocket motors, the fuel tanks, and the other parts of the rocket structure also have mass. For example on the space shuttle the two solid rocket boosters and the main fuel tank when empty have a mass of over 200 tonnes. This is more than the mass of the shuttle itself! Roughly 10% of the mass of a rocket on takeoff is "dead weight" – the mass of the empty rocket structure. But as the following calculation shows we can only get about 9.8% of a rocket's original mass into orbit.
| final mass | x 100 = % | so |
100 | x 100 = 9.78 % of original mass into orbit | ||
| original mass | 1022.67 |
In other words, we could get the rocket's structure into orbit, but no payload – no crew, and no cargo. With current technology it isn't possible to build a single stage rocket light enough, yet get enough propellant into it to get it into earth orbit with any useful payload.
The main advantage of using multi-stage rockets is that you throw away all this dead weight, and just accelerate a much lighter second stage. Since each stage doesn't need to accelerate to the final velocity, we can get our final payload into orbit.
Now repeat the above calculation, but use a two stage rocket. Assume the first
stage will accelerate the rocket to one quarter of its orbital velocity, but that it will
have all the atmospheric drag (since air resistance decreases rapidly with altitude, most
of the air resistance occurs in the first few km of travel). The v for the two stages can be calculated as
follows:
| Stage 1: | Final velocity = (7800 m/s)(0.25) | 1950 m/s | ||
| + 1500 m/s atmospheric drag | 1500 m/s | |||
| Initial velocity | 0 m/s | |||
| v |
3450 m/s | |||
| Stage 2: | Final Velocity | 7800 m/s | ||
| Initial Velocity | 1950 m/s | |||
| v |
5850 m/s | |||
Calculate the initial mass of the second stage, to achieve the required
v and put 100 tonnes of
payload and another 100 tonnes to represent the structure of the empty
second stage (total of 200 tonnes) into orbit. Use this value of mass (200 tonnes),
and the v for stage 2 (5850 m/s)
to calculate the initial mass of stage 2. Take the initial mass of stage 2, add to
it another 300 tonnes to represent the structure of empty stage 1, and calculate the
initial mass of stage 1 if it has to achieve a v of 3450 m/s.
|
With current technology we can't build a single stage rocket that would get a big enough payload into orbit. However, it is an area of active research, since it will make space travel much cheaper (imagine how much it would cost to take an airplane flight if the airplane was thrown away after every flight!)
