How do rockets attain high speeds?

[dreamLord]

The equation relating velocity of a rocket to it's mass is given by v1-v2 = uln(m1/m2), where v1, m1 and v2,m2 are masses and velocities at some times t1 and t2.

Assuming a fuel mass to rocket mass ratio of 99 : 1, we get final velocity to be only 2.3 times the initial velocity. How then do rockets attain speeds of over 10000 mph?

This is a question asked in Morin's Classical Mechanics text. Any help?

 

[russ_watters]

What initial velocity? Rockets start off stationary.

 

[dreamLord]

My mistake, obviously they start stationary - u is the velocity of the fuel mass ejected with respect to the rocket. Anyway, the question still holds.

 

[SteamKing]

Ask the Folks who are in the rocket business:
http://exploration.grc.nasa.gov/education/rocket/rktpow.html

 

[The_Duck]

The equation relating velocity of a rocket to it's mass is given by v1-v2 = uln(m1/m2), where v1, m1 and v2,m2 are masses and velocities at some times t1 and t2.

Assuming a fuel mass to rocket mass ratio of 99 : 1, we get final velocity to be only 2.3 times the initial velocity. How then do rockets attain speeds of over 10000 mph?

$u$ is the exhaust velocity, not the initial velocity. The exhaust velocity is quite large: typically several km/s. See the example numbers given for "solid rocket" and "bipropellant liquid rocket" on Wikipedia.

 

[dreamLord]

@SteamKing : I have no idea what they are trying to say! Way too many symbols floating around for me to handle.

@The_Duck : Yeah, I noticed my mistake after russ pointed it out. And thanks, I guess that explains it. Didn't bother to think of such a simple solution!

 

[Borek]

v1-v2 = uln(m1/m2)

One important point - this is for a single stage rocket.

How then do rockets attain speeds of over 10000 mph?

In several steps.

 

[dreamLord]

What do you mean by a single stage rocket? One in which we are assuming all the fuel to be released at once?

 

[Borek]

http://en.wikipedia.org/wiki/Multistage_rocket

 

[D H]

What do you mean by a single stage rocket? One in which we are assuming all the fuel to be released at once?

A single stage rocket comprises engines, fuel system, and payload, all held together within some structural housing. To go faster you need to add more fuel. At some point this becomes impossible. You can't overfill the fuel tanks. You need a bigger rocket. Now you have a new problem: That bigger rocket has more mass. The ideal rocket equation has a lot to say about the limitations of single stage rockets.

In a two stage rocket, the first stage is a big rocket whose payload is a smaller single stage rocket. That single stage rocket remains inactive until the first stage runs out of fuel. At this point, the second stage separates from the first stage and ignites. The advantage of doing this is that the second stage isn't hauling that now useless first stage around.

This nesting can be carried on ad infinitum, but with the obvious limitation that the subsequent stages get smaller and smaller. The Apollo program rockets essentially were six stage rockets. The Saturn V launch vehicle was a three stage rocket whose payload was the Apollo spacecraft proper. The Saturn V was a huge behemoth that launched the vehicle (stage 1), brought it up to low Earth orbit (stage 2), and sent the Apollo spacecraft toward the Moon. The Apollo spacecraft was itself a three stage rocket. The Service Module put the spacecraft into lunar orbit and later brought everyone home. The Lunar Excursion Module was a two stage rocket. The descent portion brought the astronauts down to the Moon. This was left behind on the Moon. The ascent portion brought the astronauts back to the Service Module for the trip home.

 

[Janus]

The equation relating velocity of a rocket to it's mass is given by v1-v2 = uln(m1/m2), where v1, m1 and v2,m2 are masses and velocities at some times t1 and t2.

Assuming a fuel mass to rocket mass ratio of 99 : 1, we get final velocity to be only 2.3 times the initial velocity. How then do rockets attain speeds of over 10000 mph?

This is a question asked in Morin's Classical Mechanics text. Any help?

A lot depends on u, the exhaust velocity.

v2 and v1 are what I assume stand for the rocket's initial velocity and its final velocity. The difference is delta v, or the change in the rocket's velocity.

The natural log of 99 is ~4.6, which means to reach 10000 mph, you would need an exhaust velocity of 2,200 mph or a little under 1000 meters per sec. The solid boosters for the Space Shuttle had an exhaust velocity of 2370 m/s. At this exhaust velocity, it would only take a mass ratio of 6.5 to reach 10,000 mph.

The Saturn 5 booster had an Ve of 2577 m/s, which would bring it down to 5.5.

But then 10,000 mph is not really fast as far as a rocket goes, its only 2.8 mps, which is well short of the 7 mps one would need to reach escape velocity. Which is why we use the multistage rockets.