Hello, and welcome to the Minute Physics tutorial on basic Rocket Science. Before we get started,
I'd like to quickly announce that I'm now making these videos at the perimeter institute for theoretical physics
So keep your eyes out for plenty of upcoming videos about string theory black holes the Higgs Boson and so on
Now, back to Rocket science. Okay, this is gonna be a super simplified version of the basic dynamics of rocket we're going to start with a
The rocket we're going to assume has some fuel in it and the whole rocket together with the fuel is going to have Mass [M]
At some point the rockets can ignite its engine start ejecting fuel and hopefully will take off
And it will have a velocity "V" in the upward direction and the exhaust will the velocity called "the velocity of the exhaust" going down
Now on top of that because we're on Earth and not in the middle [of] outer space
the Whole Rocket and
Fuel system, everything together, is going to be affected by the force of gravity. [so] [how] did the dynamics of this system work?
Well, you've probably all heard of F=Ma
Force is equal to mass times acceleration
Now in this case what it means is that the force acting on a system causes an acceleration?
Or a change of velocity of the system, and you need a bigger Force to move a bigger Mass
And that's why the m is in there
So we're going to start off by looking at the case
Before any of the fuel starts ejecting it'll just be the rocket with Mass M
So the total Force is the force of gravity and the force of gravity is equal to the mass of the object times
the gravitational acceleration
And that's equal to, of course, Ma which is just the big M again times the acceleration
And it's negative because the force of gravity is in the downward direction. Now in this case the acceleration is negative, the acceleration from Gravity
basically if the rockets in [Midair] and it's not
firing it's going to start falling down at a rate of 9.8 meters per second squared, so
after the rocket ignites not all the mass will be in the rocket anymore because some of it will be out here and the exhaust
that's been shot out. In this case the total force stays the same because it's still minus the total mass times gravity
But instead of just writing, Ma on the right side again,
we now have two separate masses, the rocket and the exhaust, so we have to add together the mass times acceleration of each
Now the mass of the rocket is "m" minus some rate of fuel loss which we'll call "R"
times the amount of time which is passed.
Basically "R" times "t" will tell you how many kilograms of fuel have shot out of the rocket is exhaust by time "t" and the
whole mass, M minus Rt is multiplied by the acceleration of the rocket.
We also have to take into account the mass from the exhaust that came out
and we just said that the exhaust comes out of the rate "r" kilograms per second, so it's mass is just "R" times "t"
So now what's the acceleration of the exhaust, well the definition of acceleration
is, "the change of velocity in a given time" and the exhaust goes from moving with the rocket at Velocity "V" before it's expelled, to
Moving with the velocity of the exhaust after it's expelled. The change or difference between those is negative
"Ve." It's negative because the exhaust is moving down minus "V" divided by the amount of time that's passed
So the "t's" cancel and this equation tells us how the whole rocket plus exhaust system moves,
where this part is the mass times acceleration of the rocket itself and this part is the mass times acceleration of the exhaust.
Now suppose you're launching a water bottle rocket or trying to levitate by vomiting
How much fuel, which in this case is just water or milk,
do you have to expel to take off, that is, to just barely beat gravity and begin to hover?
Well, if you're just hovering that means your velocity is zero and your acceleration is zero too.
So all the parts of this complicated equation having to do with the rocket go away
and we're left with a much simpler equation describing just the exhaust "m" times "g" equals "R" times "ve"
Remember, we're trying to determine how much water needs to be expelled in order for the rocket to hover but now we have two variables
which are trying to tell us that "R" is the rate at which the exhaust water leaves the rocket and
"Ve" is the speed it has after leaving; those sound pretty similar.
Let's see if we can figure out a way to relate them to each other.
On the bottom of the Rocket there's probably a circular opening with area "a" or the exhaust comes out and since liquids like milk and water are
Incompressible, they'll take up the same volume when they come out, as when they were inside the rocket.
So there will be a stream of water exhaust shooting out of the rocket and in one second, it'll go a distance
"Ve" that's the velocity of the exhaust
This stream is roughly a cylinder with the volume "a" times "ve" measured in cubic meters.
How do we relate this volume to the rate "R" of exhaust leaving the rocket?
well "R" tells us the number of kilograms of water and a kilogram of water takes up a liter of space and there are a
thousand liters in a key meter, so "R" is just 1,000 times the volume.
Now that we can relate both "R" and the velocity of the exhaust to the volume of the exhaust,
it's just simple Algebra to plug this into the equation describing the exhaust.
Do a little multiplying dividing
square rooting and we've solved for the volume. All
that remains is to plug in numbers for "m" the total mass of Rocket + fuel in
Kilograms and "a" the area of the opening at the bottom of the rocket; hint, it's probably a circle.
So what are you waiting for go weigh yourself,
measure your throat, and find out how much milk you'd have to vomit in order to levitate and
if you want you can share your personalized levitation inducing milk vomiting rate in the comments below.
Thanks for watching!