Gravitational Potential Energy Example

by Jared Rovny

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    00:01 Let's look at a quick example of gravitational potential energy.

    00:04 And suppose you dropped 3 meters into a pool.

    00:07 What was your velocity when you reached the water? So try this on your own.

    00:12 We have now an idea of the kinetic energy.

    00:14 We have an idea of the potential energy.

    00:15 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.

    00:23 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.

    00:29 Let's do that here.

    00:31 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.

    00:40 That where we're gonna call the pool zone, a height at zero meters.

    00:43 You could have solved this question which is asking you know.

    00:47 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.

    00:54 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.

    01:07 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.

    01:16 So first, we have our initial energy which is the initial kinetic energy and the initial potential energy.

    01:23 Finally, we have the final energy which is the final kinetic energy and the final potential energy.

    01:32 So again we write the energy at the two snapshots.

    01:34 What's the total energy initially, what's the total energy finally and we're going to compare.

    01:38 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.

    01:47 Remember that the kinetic energy is 1/2 mass times velocity squared.

    01:51 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.

    01:58 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.

    02:05 So no kinetic energy initially.

    02:08 Let's do the same sort of analysis for the energy finally.

    02:11 Let's look at the energy when you are at the pool.

    02:13 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.

    02:26 You are exactly at the ground.

    02:28 So we define our coordinate system, in a way that the ground or your final location will be your zero point.

    02:34 Where your, you don't have any height and therefore you don't have any potential energy.

    02:37 And so we'll just cancel this energy because it also a zero because the height is zero.

    02:42 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.

    02:53 The initial potential energy is mgh.

    02:57 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.

    03:09 We'll apply our conservation of energy and this is what's going to finish and solve this problem for us.

    03:14 By applying conservation of energy, we simply can say mgh is equal to 1/2 mv squared.

    03:23 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.

    03:36 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.

    03:49 And even here, I've written everything out in full detail with all the energy, the kinetic and potential at each location.

    03:56 We saw how quick and easy it was and once you understand that, when you are not moving you have no kinetic energy.

    04:01 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.

    04:09 For completeness, let's write out what the answer is in terms of the numbers we've been given.

    04:13 We have the square root of 2.

    04:14 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.

    04:23 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.

    04:37 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.

    04:45 For example, you know that 8 squared is 64.

    04:49 You know that 7 squared is 49 and so you know that the square root of 60 must be somewhere between 7 and 8.

    04:56 And in fact, that's exactly the case, it turns out that this is approximately 7.750 meters per second.

    05:06 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.

    05:14 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.

    05:32 This will help you solve these problems much more quickly.

    05:39 Let's take one more example of a gravitational potential energy problem.

    05:43 And let's now do a sort of the opposite of what we just did.

    05:47 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.

    05:54 So you see that we're reversing this rather than knowing the height and finding the velocity.

    05:58 We're starting with the velocity and we're going to try to find the height.

    06:02 For this sort of problem, we have exactly the picture as you see it here.

    06:05 We initially have a kinetic energy and we end up with just potential energy.

    06:09 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.

    06:24 The final energy E2 is equal to just the potential energy because right at the apex of motion the object does not moving.

    06:32 So, it's always important to know that the apex of motion, the velocity and the vertical direction is always zero.

    06:37 So we have that the final energy is just the potential energy.

    06:41 And now we get to do the exact same type of analysis, we say that E1 is equal to E2 by conservation of energy.

    06:48 E1 is ' the mass times the velocity squared that we started with.

    06:53 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.

    07:02 So say that h is equal to, dividing both sides by g, v squared over 2g.

    07:08 And so this problem isn't so difficult either we just equate the energies, one is entirely kinetic, one is entirely potential.

    07:16 And then we can plug in the values that we have.

    07:18 In this case specifically, we happen to have the velocity is 5.

    07:22 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.

    07:29 This is 25 over 2 times 10 is 20.

    07:33 So let's put an approximation here.

    07:36 So this is about 1.250 meters in height.

    07:42 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.

    07:49 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.

    08:00 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.

    08:12 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.

    08:22 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.

    08:30 So this is something you should be able to do entirely on your own very quickly.

    08:33 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.

    08:44 Just to make sure we have the right answer here, if you wanna check your work.

    08:48 We're just going to do the exact same analysis and say that the initial energy is just potential energy, mgh.

    08:54 The final energy is just kinetic energy, 1/2 mass times velocity squared.

    08:59 These energies as we've done two times now are the same.

    09:04 Mgh is 1/2 mv squared.

    09:07 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.

    09:16 So this will be exactly like the falling-into-the-pool problem.

    09:18 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.

    09:30 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.

    09:41 And so hopefully you're very comfortable now with how to solve energy problems.

    09:45 Where you know the potential energy is one thing at one point and the kinetic energy is something at a different point.

    About the Lecture

    The lecture Gravitational Potential Energy Example by Jared Rovny is from the course Energy of Point Object Systems.

    Included Quiz Questions

    1. 6J.
    2. 5.5J.
    3. 4J.
    4. 5J.
    5. 7J.
    1. 200kJ.
    2. 400kJ.
    3. 350kJ.
    4. 150kJ.
    5. 450kJ.
    1. 720J.
    2. 72.5J.
    3. 725J.
    4. 715J.
    5. 730J.

    Author of lecture Gravitational Potential Energy Example

     Jared Rovny

    Jared Rovny

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