Conservation of Energy Using the law of conservation of energy to see how potential energy is converted into kinetic energy
Conservation of Energy
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- Welcome back.
- At the end of the last video, I left you
- with a bit of a question.
- We had a situation where we had a 1 kilogram object.
- This is the 1 kilogram object, which I've drawn neater in
- this video.
- That is 1 kilogram.
- And we're on earth, and I need to mention that because
- gravity is different from planet to planet.
- But as I mentioned, I'm holding it.
- Let's say I'm holding it 10 meters above the ground.
- So this distance or this height is 10 meters.
- And we're assuming the acceleration of gravity, which
- we also write as just g, let's assume it's just 10 meters per
- second squared just for the simplicity of the math instead
- of the 9.8.
- So what we learned in the last video is that the potential
- energy in this situation, the potential energy, which equals
- m times g times h is equal to the mass is 1 kilogram times
- the acceleration of gravity, which is 10
- meters per second squared.
- I'm not going to write the units down just to save space,
- although you should do this when you do it on your test.
- And then the height is 10 meters.
- And the units, if you work them all out, it's in newton
- meters or joules and so it's equal to 100 joules.
- That's the potential energy when I'm holding it up there.
- And I asked you, well when I let go, what happens?
- Well the block obviously will start falling.
- And not only falling, it will start accelerating to the
- ground at 10 meters per second squared roughly.
- And right before it hits the ground-- let me draw that in
- brown for ground-- right before the object hits the
- ground or actually right when it hits the ground, what will
- be the potential energy of the object?
- Well it has no height, right?
- Potential energy is mgh.
- The mass and the acceleration of gravity stay the same, but
- the height is 0.
- So they're all multiplied by each other.
- So down here, the potential energy is going
- to be equal to 0.
- And I told you in the last video that we have the law of
- conservation of energy.
- That energy is conserved.
- It cannot be created or destroyed.
- It can just be converted from one form to another.
- But I'm just showing you, this object had 100 joules of
- energy or, in this case,
- gravitational potential energy.
- And down here, it has no energy.
- Or at least it has no gravitational potential
- energy, and that's the key.
- That gravitational potential energy was converted into
- something else.
- And that something else it was converted
- into is kinetic energy.
- And in this case, since it has no potential energy, all of
- that previous potential energy, all of this 100 joules
- that it has up here is now going to be converted into
- kinetic energy.
- And we can use that information to figure out its
- velocity right before it hits the ground.
- So how do we do that?
- Well what's the formula for kinetic energy?
- And we solved it two videos ago, and hopefully it
- shouldn't be too much of a mystery to you.
- It's something good to memorize, but it's also good
- to know how we got it and go back two videos if you forgot.
- So first we know that all the potential energy was converted
- into kinetic energy.
- We had 100 joules of potential energy, so we're still going
- to have 100 joules, but now all of it's going to be
- kinetic energy.
- And kinetic energy is 1/2 mv squared.
- So we know that 1/2 mv squared, or the kinetic
- energy, is now going to equal 100 joules.
- What's the mass?
- The mass is 1.
- And we can solve for v now.
- 1/2 v squared equals 100 joules, and v
- squared is equal to 200.
- And then we get v is equal to square root of 200, which is
- something over 14.
- We can get the exact number.
- Let's see, 200 square root, 14.1 roughly.
- The velocity is going to be 14.1 meters per
- second squared downwards.
- Right before the object touches the ground.
- Right before it touches the ground.
- And you might say, well Sal that's nice and everything.
- We learned a little bit about energy.
- I could have solved that or hopefully you could have
- solved that problem just using your kinematics formula.
- So what's the whole point of introducing
- these concepts of energy?
- And I will now show you.
- So let's say they have the same 1 kilogram object up here
- and it's 10 meters in the air, but I'm going to change things
- a little bit.
- Let me see if I can competently erase all of this.
- Nope, that's not what I wanted to do.
- OK, there you go.
- I'm trying my best to erase this, all of this stuff.
- So I have the same object.
- It's still 10 meters in the air and I'll
- write that in a second.
- And I'm just holding it there and I'm still going to drop
- it, but something interesting is going to happen.
- Instead of it going straight down, it's actually going to
- drop on this ramp of ice.
- The ice has lumps on it.
- And then this is the bottom.
- This is the ground down here.
- This is the ground.
- So what's going to happen this time?
- I'm still 10 meters in the air, so let me draw that.
- That's still 10 meters.
- I should switch colors just so not everything is ice.
- So that's still 10 meters, but instead of the object going
- straight down now, it's going to go down here and then start
- sliding, right?
- It's going to go sliding along this hill.
- And then at this point it's going to be going really fast
- in the horizontal direction.
- And right now we don't know how fast.
- And just using our kinematics formula, this would have been
- a really tough formula.
- This would have been difficult.
- I mean you could have attempted it and it actually
- would have taken calculus because the angle of the slope
- changes continuously.
- We don't even know the formula for the angle of the slope.
- You would have had to break it out into vectors.
- You would have to do all sorts of complicated things.
- This would have been a nearly impossible problem.
- But using energy, we can actually figure out what the
- velocity of this object is at this point.
- And we use the same idea.
- Here we have 100 joules of potential energy.
- We just figured that out.
- Down here, what's the height above the ground?
- Well the height is 0.
- So all the potential energy has disappeared.
- And just like in the previous situation, all of the
- potential energy is now converted into kinetic energy.
- And so what is that kinetic energy going to equal?
- It's going to be equal to the initial potential energy.
- So here the kinetic energy is equal to 100 joules.
- And that equals 1/2 mv squared, just
- like we just solved.
- And if you solve for v, the mass is 1 kilogram.
- So the velocity in the horizontal direction will be,
- if you solve for it, 14.1 meters per second.
- Instead of going straight down, now it's going to be
- going in the horizontal to the right.
- And the reason why I said it was ice is because I wanted
- this to be frictionless and I didn't want any energy lost to
- heat or anything like that.
- And you might say OK Sal, that's kind of interesting.
- And you kind of got the same number for the velocity than
- if I just dropped the object straight down.
- And that's interesting.
- But what else can this do for me?
- And this is where it's really cool.
- Not only can I figure out the velocity when all of the
- potential energy has disappeared, but I can figure
- out the velocity of any point-- and this is
- fascinating-- along this slide.
- So let's say when the box is sliding down here, so let's
- say the box is at this point.
- It changes colors too as it falls.
- So this is the 1 kilogram box, right?
- It falls and it slides down here.
- And let's say at this point it's height above the ground
- is 5 meters.
- So what's its potential energy here?
- So let's just write something.
- All of the energy is conserved, right?
- So the initial potential energy plus the initial
- kinetic energy is equal to the final potential energy plus
- the final kinetic energy.
- I'm just saying energy is conserved here.
- Up here, what's the initial total energy in the system?
- Well the potential energy is 100 and the kinetic energy is
- 0 because it's stationary.
- I haven't dropped it.
- I haven't let go of it yet.
- It's just stationary.
- So the initial energy is going to be equal to 100 joules.
- That's cause this is 0 and this is 100.
- So the initial energy is 100 joules.
- At this point right here, what's the potential energy?
- Well we're 5 meters up, so mass times
- gravity times height.
- Mass is 1, times gravity, 10 meters per second squared.
- Times height, times 5.
- So it's 50 joules.
- That's our potential energy at this point.
- And then we must have some kinetic energy with the
- velocity going roughly in that direction.
- Plus our kinetic energy at this point.
- And we know that no energy was destroyed.
- It's just converted.
- So we know the total energy still has to be 100 joules.
- So essentially what happened, and if we solve for this--
- it's very easy, subtract 50 from both sides-- we know that
- the kinetic energy is now also going to
- be equal to 50 joules.
- So what happened?
- Halfway down, essentially half of the potential energy got
- converted to kinetic energy.
- And we can use this information that the kinetic
- energy is 50 joules to figure out the
- velocity at this point.
- 1/2 mv squared is equal to 50.
- The mass is 1.
- Multiply both sides by 2.
- You get v squared is equal to 100.
- The velocity is 10 meters per second along
- this crazy, icy slide.
- And that is something that I would have challenged you to
- solve using traditional kinematics formulas,
- especially considering that we don't know really much about
- the surface of this slide.
- And even if we did, that would have been a million times
- harder than just using the law of conservation of energy and
- realizing that at this point, half the potential energy is
- now kinetic energy and it's going along the
- direction of the slide.
- I will see you in the next video.
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