A rocket for use in deep space is to have the capability of...

Key Concepts

Tsiolkovsky Rocket Equation

This fundamental equation in rocketry relates the change in velocity of a rocket to its exhaust velocity and the ratio of its initial mass (including fuel) to its final mass (after fuel burn). Expressed as ?v = v? ln(m?/m_f), it is essential for determining how much fuel is needed to achieve a given velocity change.

Exhaust Velocity

Exhaust velocity is the speed at which propellant is expelled from the engine relative to the rocket. It directly influences the efficiency of the propulsion system; a higher exhaust velocity allows the rocket to attain a greater change in velocity for a given amount of propellant, according to the Tsiolkovsky rocket equation.

Mass Ratio

The mass ratio is the quotient of the rocket’s total initial mass (including fuel, oxidizer, and structure) and its dry mass (after fuel burn). It is a crucial factor in rocket design because the required fuel mass to achieve a certain velocity change depends logarithmically on this ratio, making it a key parameter in determining the feasibility of a mission.

Conservation of Momentum

This physical principle underlies the operation of rocket propulsion. It dictates that the momentum gained by the rocket is equal to the momentum carried away by the ejected propellant, thereby providing the thrust needed to accelerate the vehicle. This concept is integral to understanding how rockets work and is embodied in the derivation of the rocket equation.

Transcript

00:01 This problem is all about rockets and figuring out how much fuel is needed to get a rocket to a certain velocity.

00:11 And we know how the change in velocity of the rocket is related to the change in mass of the rocket, because the rocket will accelerate because it is throwing mass out behind it through the expulsion of fuel.

00:31 And so the mass is going to change.

00:34 So as the mass changes, the velocity is going to increase.

00:38 The mass changes also because we are, as we throw fuel out of the rocket, we're losing mass constantly.

00:47 And so we know how to relate the change in velocity of the rocket to the exhaust velocity, which is the velocity of the exhausted fuel relative to the rocket.

00:57 And the initial mass, which is the mass of the rocket and fuel before, or when it's at the initial velocity, and the final mass is the mass of the rocket and the fuel when it's at the final velocity.

01:18 This was one of the results that was given in the book.

01:22 If we're given that the mass of the rocket, which is the mass of the mass, which is the mass, of the frame, the rocket frame, the engine, which will be somewhere in the rocket, and the payload, which the payload is just the whatever the rocket is carrying, whatever it wants to bring in this space, which could be a satellite.

01:48 We're given that the mass of all of these things, but not the fuel, just the mass of the rocket, is three metric tons.

02:01 And we want to find out what the mass of the fuel is when the, we'll start with an exhaust velocity of 2 ,000 meters per second, and we want to know how much fuel is required to get to the rocket to a velocity of 10 ,000 meters per second, which i will label as just v.

02:32 If we assume that the rocket starts at rest, which we are, we're going to assume that this final velocity is after takeoff.

02:43 So the initial velocity is when the rocket is still on earth sitting in the space station.

02:50 So v .i will be zero meters per second.

02:55 So if we take all of this into account and we take this rocket equation, vf, i'm going to keep in general, so vf minus v i, v i is zero.

03:07 So this is the same thing as just vf is equal to the exhaust velocity times the natural log of the initial mass over the final mass.

03:21 So we made one simplification.

03:24 Now we said that the initial mass m sub i is the initial mass of the rocket and we're going to assume the mass of the rocket does not change.

03:37 So the frame, the engine, and the payload, all those things stay intact.

03:43 Plus, this is plus the mass of the fuel initially.

03:48 So we have fuel in the tank.

03:51 However much fuel we have in the tank also contributes to this initial mass.

03:56 The final mass is the mass of the rocket when it gets to this 10 ,000 meters per second, plus the mass of the fuel afterwards.

04:11 And since we want to know the amount of fuel to get to this velocity, if we had any more fuel, we would burn it and thus get faster.

04:23 So we want to know the amount of fuel to get to this exact velocity.

04:28 So when we're at this velocity, we can't have any more fuel.

04:32 So the velocity at the final location when it gets the 10 ,000 meters per second is zero.

04:37 We have no more fuel.

04:38 So that means the final mass will just be the mass of the rocket.

04:45 And we were given that the mass of the rocket is three metric tons, and i'll just label that as three, and we can remember the units.

04:52 So the initial mass, we then know, is three metric tons, plus the mass of the fuel, how much fuel we need to get to this velocity, which is what we're trying to solve for.

05:05 So then we want to arrange this equation.

05:08 We can also write it out in terms of what we just found.

05:13 The exhaust velocity times the natural log of the initial mass we found is just the mass of the rocket.

05:20 I'll just label that as m sub r plus the mass of the fuel divided by the final mass, which is just the mass of the rocket.

05:32 And we want to solve this for the mass of the fuel.

05:34 We know everything.

05:36 We know the mass of the rocket...