This problem is a challenge problem and it's not a challenge problem because it's particularly
mathematically difficult, but rather the concept is really tricky to understand.
We're going to be talking about escape velocity.
What is escape velocity? Well to understand this let's imagine a rocket on the surface of the earth.
First let's imagine that this rocket takes off with a very small speed. It's not going very fast.
Then you can imagine that gravity pulling on the ship will cause it to slow down
and fall back down to earth.
Even at greater speed it might take off, finally come to a stop, and then fall back towards the earth.
And this makes sense because the force of gravity is always pulling on the rocket.
In fact, no matter how far away the rocket is from the earth,
the earth will always be pulling on the rocket.
If we draw a force diagram for a rocket, we see that the only force acting on it
is the force of gravity pulling it back towards earth.
No matter what speed the rocket is going, it's always accelerating back towards earth.
And since it's accelerating back towards earth, the rocket is always slowing down.
It makes sense that at some point it comes to rest and then falls back to earth.
Even though this rocket is always accelerating towards earth and in fact is always slowing down
it actually is still possible for the rocket to escape the earth forever
and never be pulled back down.
Just keep going away from the earth forever. That seems like a strange idea.
If it's always slowing down, how can it avoid stopping completely? Well, let's find out.
The answer to this question actually lies in this formula for
gravitational potential energy that we just discussed.
Now if you remember our discussion of potential energy in the unit,
we discussed how we can always set our potential energy to be 0
at any point we want as long as we stick to it.
In fact this formula could have some constant added onto it
to make it so the gravitational was 0 at a specific location.
Let's be smart in picking a good choice for what we want the zero potential energy point to be.
I'm going to choose a point really, really, really far away from the earth
to be where my gravitational potential energy is equal to 0.
And it just so happens that this choice for a potential energy of zero point
happens to be when K equals 0 and let's see why.
As we get further and further away from the earth,
R, the distance from the rocket to the earth, gets bigger and bigger and bigger
which means this value continues to get closer to 0.
And the only time this value ever gets to be 0 is when R goes to infinity
at some point really far away from the earth.
That's why we don't actually need that constant K here to get that same effect.
Okay. But that doesn't really explain why I chose the zero point to be off at infinity.
Why wouldn't I choose it at like the surface of the earth or something else like that.?
Well let's find out how choosing our zero point for gravitational potential energy at infinity
helps us reason about escape velocity.
Now, once again, we're going to tackle this problem by using conservation of energy.
The idea that the sum of the kinetic and the potential energy
is the same at any point in the problem.
For my starting point, I'm going to choose the original rocket taking off point.
At which point, the rocket has some initial kinetic energy and some initial potential energy.
And for my final point, I'm going to choose the point when the rockets finally stops moving
the acceleration caused by the force pulling it back towards the earth
finally causes its velocity to reach 0.
At this point, there is no kinetic energy.
All right. So now we want to consider a few cases of what happened if the sum
of the kinetic and potential energy when the rocket lifts off has different values.
First, I want to consider the case where the sum is less than 0.
If the kinetic energy plus the original potential energy is less than 0
then we know that the final potential energy must also be less than 0.
And this is great because if we look at our general formula for gravitational potential energy
we see that the value is always negative because of this negative sign on the front.
We should be easily able to solve for R,
the distance the rocket is away from the earth when its speed finally reaches 0.
Now, what about the case where the original kinetic energy
plus the original potential energy is greater than 0?
Well then in this case the final potential energy must also be greater than 0.
But this poses a problem since this formula says that gravitational potential energy
is always negative, we're in trouble.
How can the final potential energy be greater than 0 and less than 0 at the same time?
The solution is they can't be. So what went wrong?
Well, to get to this point, we assumed that the final kinetic energy was 0.
If setting the final kinetic energy equal to 0 leads to a contradiction,
then it must be the case that the final kinetic energy can't ever be 0.
And what does that mean? It means that the rocket never slows down to 0 velocity.
Even though it's always accelerating towards the earth,
it never slows all the way down and falls back to earth.
If the rocket never reaches 0 velocity and never falls back to earth,
we can say that the rocket has successfully escaped.
And that's what it means for the rocket to have escape velocity.
Its original kinetic energy plus the original potential energy is greater than 0.
And as you might've guessed, the threshold case where the rocket
has just enough velocity to escape is when the kinetic energy
plus the potential energy is exactly equal to 0.
Awesome. All right. Now that you know this fact, you're ready to tell me
what speed does the rocket need to be going in order to escape the earth
from the surface of the earth? What is its escape velocity?
In order to solve this problem you're going to need these constants.
The value of big G, the value of the radius of the earth, and the mass of the earth.
Good luck finding the escape velocity.
And if you're confused about any of the topics I just discussed,
please head to the forums and ask your questions there. I'll be happy to answer them.