 In this video, I'm going to develop the rocket equation based on linear momentum and hopefully I will be able to show you that rocket science actually is not that difficult. So let's start with the situation you're having. We are having a rocket of mass m, which includes the fuel that is stored within the rocket. This rocket is traveling in space at initial velocity v. There are particles leaving the rocket at the back at the relative speed to the rocket of v rel. And these particles as measured from somebody outside the rocket that would not be traveling would be traveling at a speed of v e. Now the first problem is how can we actually figure out what v e is. v e, the speed of an exhaust particle that somebody at the ground would measure, is simply the speed of the rocket minus the relative speed. So once we have this, we have our basic setup. The next question is what is our initial momentum? At the beginning, when the rocket is traveling, it's only the rocket that has momentum which is equal to the total mass of the rocket and the fuel times the velocity of the rocket. In our x direction, this means if I remove the vectors, I have a positive initial momentum which is equal to m v. The next question now is what is the momentum after some particles are shot out, so my final momentum. My final momentum is equal to the rocket that's now traveling at the new speed, that's v final. But the mass of the rocket now went down a bit, it went down by the amount of particles that left the rocket on the back, that delta mass. However, our system still includes the particles that are flying off, so we have to include their momentum as well. So we have plus change in momentum, sorry not change in momentum, the mass of these particles flying at my exhaust speed. I'm now going to rewrite this equation without the vector heads and only in x direction and at the same time I will replace my exhaust velocity by what I've already determined at the beginning here. And with the two velocities that are already knocked, so my p final, I'm also trying to the right, so positive number will be the mass of the rocket minus the mass that left the rocket times the final speed of the rocket. Plus that little mass that left the rocket times the velocity of the rocket minus the relative speed of the rocket. At that moment, my velocity of the rocket is actually the same as my final velocity of the rocket, I'm looking at over here. Therefore, I can write this one as the final. Now, looking at this, I actually see that some of my terms are cancelling out, so if I go here, if I multiply it out, I have mass times v final minus delta m v final plus delta m v final minus delta m relative speed. So these two cancel out. Therefore, I can rewrite my final momentum as mass v final minus change of mass v relative. And that should be equal to my initial momentum that I found here. So you have total mass times initial velocity is total mass, hence fine velocity minus change of mass times v relative, the speed of the exhaust. Now my v final is nothing else than my initial velocity plus a little change in velocity, how much it went faster. I need some space on my board, so I'm going to try if I can move something a bit away. So my final momentum we had here and all of this goes here. So all I want to do now is I actually multiply this out. So I have my initial momentum is equal to my final momentum plus m delta v minus delta m v relative. And what you should see is that my m delta v actually cancels out. So what I get if I rearrange this a bit I get delta m v relative is equal to mass delta v. It really gets interesting if I take this last equation here and I divide by the time. That little bit of time that passed between my initial and my final moment. So if I take delta m over delta t v relative is m delta v over delta t. Now what I just found is this one here, this is my acceleration, velocity change of velocity over time. And this one here, this is my rate of mass loss, how much mass, how many kilograms per second I'm losing. So what I just found is r v relative is the total mass of the rocket times the acceleration. And this is the first rocket equation. So the faster something is leaving the rocket the more acceleration I will get. The more material is leaving the records per second the more acceleration I get. And then however how heavier the rocket including the fuel is the less acceleration I will get. And now this is our rocket science.