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Equations of Motion: Example

by Jared Rovny
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    00:01 Here's an example of this, we're going to walk through a good equations of motion example.

    00:04 Supposed you drop a penny from the top of a very tall building, in this case, a 1000 meter tall building, people are often asking, is this dangerous? Am I gonna hurt anybody if I drop a small object from a high distance like this? We are going to ignore air resistance which really would turn out to be pivotal but we haven't gotten there yet.

    00:20 And we can ask how fast will this object be going when it reaches the ground.

    00:23 We're going to assumed a constant acceleration downwards of 9.8 meters per second squared which as we will see as the gravitational acceleration for earth.

    00:32 We'll get to the physics of that later, but right now you can assume that you know the constant acceleration.

    00:36 Let's see what this looks like, but again I recommend you, you give this a shot.

    00:40 You know the acceleration, you know the equations of motion, see if you can find the final velocity of this object right before it hits the ground.

    00:48 If you've given this problem a shot, let's go through what it looks like in practice.

    00:53 First, we've drawn a nice picture here of exactly what this situation would look like, we have a penny or some small coin or object that you start at a very high height, you let go and it falls to the ground.

    01:05 In this problem, we can identify pretty quickly what our initial and final positions are, our initial position, as you can see in our second step cataloging the things that we know from the problem is a thousand meters in the air that's our initial position or x nought.

    01:19 The initial velocity, because it tells us the object was dropped is zero meters per second.

    01:24 And this is what I was talking about when I said, you might need to infer some quantities from the problem.

    01:30 Because nowhere in the problem statement was I told that the velocity of this object is zero.

    01:35 However, the fact that I know it was dropped, tells me that it must have been moving when I let go.

    01:41 Finally, as given to us in this problem the acceleration of this objects fells is 9.8 meters per second squared.

    01:47 We put a negative there because the direction of motion is downwards, in other words, the acceleration is trying to make your velocity and your position less and less and less as time goes on.

    01:56 Now let's catalog our final, so you can see we have our snapshot, our initial and our final, we have a penny at a great height and then we have it on the ground.

    02:04 And finally, our final position is zero meters, its right on the ground, you can see in this coordinates system I've introduced, that the ground is what I'm calling zero meters, that's my origin and then I measure any distances from that.

    02:17 Moving forward, we can see that the final velocity is exactly what we're trying to find, so this would be a great way to start cataloging the things that we don't know, and this would be a great place to start and just solving this problem.

    02:28 But just for completeness let's also introduced the final acceleration which is simply the same as the initial acceleration, because in this problem, and if most equations of motion problems since the equations of motion rely on constant acceleration we'll be assuming a constant acceleration as you see here.

    02:45 So now, in solving this problem we're going try to start cataloging all things that we know and don't know and find their ways start working our ways towards a solution.

    02:54 I've listed here all the equations of motion, the question is if I'm trying to find the final velocity, which one should I used? I have all these variables floating around in 3 whole equations I could sit here and start plugging numbers into, but that would leave me in a bind, so we wanted to do this in a very motivated way and say which one should I use.

    03:12 If you look at your 3 equations and you look at the things that we know and don't know you might notice one thing that we do not know is the time.

    03:19 Nowhere in my catalog, I have listed the time, you don't see the letter T anywhere, in other words, we don't have any idea how long this penny takes to hit the ground, we just know some things about, its initial starting point and its final position.

    03:30 That means that we should probably not use this first two equations, they have time in them, that would leave us sort of in a bind as I talked about rewriting equations over and over, so we really wanna stick with this last equation, because this last equation, has things that we know including the initial velocity, the acceleration, both the initial and final positions, but also has the things that we don't know, the final velocity which what we want to solve for, and that's great for us because we just want to rearrange this equation and trying to find what that final velocity is.

    03:58 Rearranging it, we can move the initial velocity to the other side by adding v nought squared to both sides, and then we can start plugging in numbers. I'm not gonna write all this out here, but if you plugged in the initial velocity of zero plus 2 times the acceleration you plugged in minus 9.8, and then you plug in your distance x minus x nought, where x is zero, and x nought is 1.000. Notice very importantly that will give you a negative number.

    04:26 A zero minus 1.000 will give you a negative 1.000 and that's also good, because if we multiply that negative 1.000 by the negative 9.8, we get the number you see here, 19600 is the final velocity squared.

    04:39 We can complete this problem by simply taking the square root of both sides and if you do so, you'll get a number which is 140 but you seen I have done something tricky here as I put in a minus sign after I took the square root, and why in the world that I do that? When you are squaring a number, we're taking the square root of a number it can be either positive or negative, because if I square a positive or square a negative I end up with a positive number either way.

    05:02 So, I need time retaking a square root of a number, you have to decide for yourself was it originally a positive number or negative number before I squared it.

    05:10 In this case, we can tell base on the physics of the problem that the number has to, have been a negative number, because our object is moving downwards and again in our coordinates system starting at zero and going to 1.000, if I'm moving downwards, I'm moving in the negative direction, I'm moving towards more negative numbers and so having solve for the number, 140 were there square root, we can introduce the negative sign by picking it from positive or negative of our square root, which is a convention.

    05:40 And remember that you can always pick your own coordinate system.


    About the Lecture

    The lecture Equations of Motion: Example by Jared Rovny is from the course Translational Motion.


    Author of lecture Equations of Motion: Example

     Jared Rovny

    Jared Rovny


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