Let's look at a quick example of gravitational potential energy.
And suppose you dropped 3 meters into a pool.
What was your velocity when you reached the water?
So try this on your own.
We have now an idea of the kinetic energy.
We have an idea of the potential energy.
You can tell me how much potential energy you have
if you're 3 meters in the air
and you also know that when you fall in,
you will have the kinetic energy.
So see if you can use all of those facts
to come up with an answer for how fast you're going,
what your velocity is when you reached the water.
Let's do that here.
Notice that in this kind of problem,
where you start at some height of 3 meters
and then you fall all the way to the ground.
That where we're gonna call the pool zone,
a height at zero meters.
You could have solved this question which is asking you know.
What is your velocity, right at the end here?
You could have solved this just using the equations of motion
because we know the gravitational acceleration, g.
And we could also just use the equations of motion
as we've been given them and solve for v,
but we also saw that that was sort of a tedious process
to write out all the equations of motion
and figure out which one we need
and decide whether we need to use units of time or not.
So we're gonna find a much more efficient way
to solve problems like this just by using energy
which is exciting because it saves us a lot of time
and it's a more intuitive concept.
So first, we have our initial energy
which is the initial kinetic energy and the initial potential energy.
Finally, we have the final energy
which is the final kinetic energy and the final potential energy.
So again we write the energy at the two snapshots.
What's the total energy initially,
what's the total energy finally and we're going to compare.
So first, notice that right when you are at the top,
since you dropped 3 meters into the pool,
you have no kinetic energy because you have no velocity.
Remember that the kinetic energy is 1/2 mass times velocity squared.
So, if you're not moving, if your velocity is zero
as it is in any case where you're dropping an object
or where you yourself in this case are dropping into a pool.
If you are not moving and you have zero velocity,
you have zero kinetic energy
and so we are going to say that this is zero.
So no kinetic energy initially.
Let's do the same sort of analysis for the energy finally.
Let's look at the energy when you are at the pool.
Notice that the potential energy,
remember this is mgh where h is the height that you are
away from your coordinate system is equal to zero as well
because you are not raised above the ground.
You are exactly at the ground.
So we define our coordinate system,
in a way that the ground
or your final location will be your zero point.
Where your, you don't have any height
and therefore you don't have any potential energy.
And so we'll just cancel this energy
because it also a zero because the height is zero.
So now, we have a nice, simple situation,
we say that you started with potential energy,
you finished with just kinetic energy
and we have expressions for both of these quantities.
The initial potential energy is mgh.
The final kinetic energy is 1/2 your mass times your velocity squared
and now we do the important thing
that I said we'll always do in these problems
which is having written that with the energy is at each snapshot.
We'll apply our conservation of energy
and this is what's going to finish and solve this problem for us.
By applying conservation of energy,
we simply can say mgh is equal to 1/2 mv squared.
We can then cancel the masses
and solve for v by multiplying both sides by 2
and taking the square root
so that we see the velocity is square root of 2 times g times h.
You might recognize this a result
from something that we did with the equations of motion
and we did the exact same sort of question
where we wanted to find the velocity
and then when we did that, we also got an answer
of the square root of 2gh, using a much longer process.
And even here, I've written everything out in full detail
with all the energy, the kinetic and potential at each location.
We saw how quick and easy it was
and once you understand that,
when you are not moving you have no kinetic energy.
When you are at the ground
you have no potential energy,
you'll be able to do that very quickly
and this whole process will be sped up
much more than it was for the equations of motion.
For completeness, let's write out what the answer is
in terms of the numbers we've been given.
We have the square root of 2.
And we can use 9.8 or 10 here
and then again, in the setting of trying to do an exam
I wouldn't be worrying about what the square root of 9.8 is.
It's possibly not a good use of your time
and then we have 2 times 3, is 6, times 10 is 60
and so our velocity would be the square root of 60
and again you could use a calculator
when you're trying to figure out,
what this exact number is, in the case of a problem.
If you wanted to find something approximate
because you didn't have a calculator on hand
but you had some options that you could choose between,
you could always do some approximation methods.
For example, you know that 8 squared is 64.
You know that 7 squared is 49
and so you know that the square root of 60
must be somewhere between 7 and 8.
And in fact, that's exactly the case,
it turns out that this is approximately 7.750 meters per second.
But again, something we want to always emphasize in these problems,
is that as you are going through you should be careful,
to use your time wisely.
If you're given a situation
where you have a square root of 60,
rather than really agonizing about where the decimal points are,
usually you'll have some options available to you
so you should look at those options
and compare them with very practical methods
like I just showed with 7 squared and 8 squared
and see what you know your answer should lie around or close to.
This will help you solve these problems much more quickly.
Let's take one more example
of a gravitational potential energy problem.
And let's now do a sort of the opposite of what we just did.
Instead of having something fall, we're going to say,
throw an apple upwards at 5 meters per second
and ask how high does it go.
So you see that we're reversing this
rather than knowing the height and finding the velocity.
We're starting with the velocity
and we're going to try to find the height.
For this sort of problem, we have exactly
the picture as you see it here.
We initially have a kinetic energy
and we end up with just potential energy.
So writing this out,
we can again say that the energy 1,
the initial energy, is just kinetic energy
because the potential energy mgh is zero
because we have no height.
The final energy E2 is equal to just the potential energy
because right at the apex of motion the object does not moving.
So, it's always important to know that the apex of motion,
the velocity and the vertical direction is always zero.
So we have that the final energy is just the potential energy.
And now we get to do the exact same type of analysis,
we say that E1 is equal to E2 by conservation of energy.
E1 is ' the mass times the velocity squared that we started with.
This will be equal to mass times g times h
and so again the mass is cancel
but this time we're gonna do something slightly different
which is to find h instead of v.
So say that h is equal to,
dividing both sides by g, v squared over 2g.
And so this problem isn't so difficult either
we just equate the energies,
one is entirely kinetic, one is entirely potential.
And then we can plug in the values that we have.
In this case specifically, we happen to have the velocity is 5.
So we have 5 squared over 2 and again this would be 9.810
00:07:26.838 --> 00:07:28.941
or approximately 10 for our problem.
This is 25 over 2 times 10 is 20.
So let's put an approximation here.
So this is about 1.250 meters in height.
What I'd recommend you to do as well is for some of these problems,
while we're making approximations is try them on your own
not making those approximations.
See what happens if you plug in the actual numbers
and all their messy detail and using a calculator
and see how close you get
and I think the more you see how close you can get to the right answer
using the approximations that we've talked about.
The more you will be convinced to the utility of a method
where you make reasonable assumptions
and then always test those assumptions
and make sure you keep track of them
as you go through a problem
in case you need to make things more accurate as you go.
So with this problem we found the height
again by making these assumptions
that the energy at one point was purely kinetic
and the energy at another point was purely potential energy.
Just for completeness, we have a similar example
where you drop an apple 5 meters
and we ask what's its velocity is when it hits the ground.
So this is something you should be able to do
entirely on your own very quickly.
So give it a shot and see if you can find
what the velocity of this apple is when it hits the ground
knowing again that it starts at a particular height 5 meters,
and then it moves all the way to the ground.
Just to make sure we have the right answer here,
if you wanna check your work.
We're just going to do the exact same analysis
and say that the initial energy is just potential energy, mgh.
The final energy is just kinetic energy,
1/2 mass times velocity squared.
These energies as we've done two times now are the same.
Mgh is 1/2 mv squared.
We can cancel these masses
and solve for the velocity as the square root of 2 times g times h,
and then plug in the values that we were given.
So this will be exactly like the falling-into-the-pool problem.
So you should get something that is approximately
the square root of 2 times 10 times 5,
which in this case is the square root of a hundred,
2 times 5 is 10, times 10 is a hundred.
So you should get something that was close to 10 meters per second
or slightly less if you're using a gravitational value
that's exactly 9.8 meters per second squared.
And so hopefully you're very comfortable now
with how to solve energy problems.
Where you know the potential energy is one thing at one point
and the kinetic energy is something at a different point.