Equations of Motion in 1 Dimension

by Jared Rovny

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    00:01 What we're going to do now is explain and introduce the equations of motion.

    00:07 I've talked a little bit about velocity as an acceleration, and I sometimes assume what the velocity would be across given time interval.

    00:14 But what we're gonna do in this case is show you how I got those numbers.

    00:18 How I got the 1.750 or the 1.5.

    00:20 And these equations are going to be pretty intuitive if you think about them, and so it's best to remember the concept instead of trying to remember all the variables inside of them.

    00:28 These equations are going to assume, very importantly, that the acceleration is constant through any interval that we're discussing.

    00:35 So if the acceleration is constant, we can ask about the velocity and the position under a constant acceleration. And they look like this.

    00:43 The velocity after a given amount of time, will be whatever it started at, v nought, the original velocity plus the acceleration times the time.

    00:51 So you can see intuitively sort of what this is telling you, the velocity will be whatever it started at, plus however much you changed it.

    00:58 And acceleration is telling you how much you changed it.

    01:01 You need to multiple by the time first to get the units right, because we solved the difference in units between acceleration and velocity, but also because if you're accelerating for a longer period of time, you're building up more and more velocity as you go.

    01:13 And so this is a pretty intuitive equation hopefully for what velocity is after a given period of time.

    01:18 The equation for position will look very similar.

    01:21 It's whatever initial position you started at plus any changes in your position given by the initial velocity times the time.

    01:29 So you see that the velocity is changing the position in the exact same way that the acceleration is changing the velocity.

    01:35 But in position we have this extra term, a plus 1/2 a t squared.

    01:40 If you're familiar with calculus you can always derive and see where this term comes from.

    01:44 But in this context all you really need to know is that if you look at the velocity that I have in this position equation, I've used v nought, just the initial velocity.

    01:53 And it might make sense intuitively that if an object is accelerating and the velocity is changing away from v nought to a new velocity, that term won't be sufficient to capture the entire motion of this object through space.

    02:03 We need a term that will account for the fact that this object might be accelerating and not just moving.

    02:08 Finally, even though these are 3 completely self-sufficient equations described in the acceleration, the velocity, and the position, we can use these 2 equations together, these final 2, the velocity and the position, to derive a 3rd equation, which says that the final velocity squared minus the initial velocity squared is twice the acceleration times the total distance you've travelled which is as you see, final minus initial, x minus x nought. Now why is this 3rd equation so useful? You notice if you look at it, it doesn't have time in it anywhere.

    02:39 And in many problems, we are trying to solve for a variable that isn't time or we don't know what the time is and so we need to use an equation that doesn't have time in it.

    02:48 And so when that happens this equation can be very, very useful.

    02:51 So these are the equations of motion.

    02:52 I highly recommend that you understand these equations in terms of what they're saying and you might want to even memorize exactly their forms because they come up very, very often in Physics context and especially in Physics problems on exams or other settings.

    03:06 If we're going to solve an equations of motion problem, like the ones that I just talked about with when introducing the equations, it usually goes something like this: First, you start by drawing a picture. I always recommend in any Physics problem that you have a good picture of what's going on if you need to draw, I would almost always do so, I certainly do so and it's highly recommended.

    03:27 This helps you contextualize and to keep things all orderly because you won't be able to hold all of the information in your head all the time.

    03:33 Second, catalog what you know.

    03:36 A lot of what you know will be given to you very specifically in the problem.

    03:40 They will say, this is this and this is that.

    03:42 But many things as we'll see are things you can't infer from the problem.

    03:46 You can figure out that this must be true or that must be true based on the physical things that have been given to me in this problem.

    03:53 Third, based on what the problem is asking you for, you should catalog what you need.

    03:57 Because as you work through a problem and are solving and changing your equations, you wanna have a very clear direction otherwise you might just be rearranging equations over and over and not really know where you're headed.

    04:08 So I recommend highly that you write down exactly what it is that you're looking for.

    04:11 Once you've written down what you know and what you need, you can catalog what are your knowns and all your unknowns.

    04:17 Because at this point you've written down many equations, you have a big picture, you have an idea of what's going on, but it can be very easy to forget or lose track of which of these variables are things that you know already and which are the things you don't know.

    04:29 And that can again leave you trying to rearrange problems over, the equations rather, over and over again.

    04:35 You don't want to find yourself in that trap because it would lose you a lot of time, and lose you directions as well. So be careful to identify what do I know in this problem and what don't I know in this problem.

    04:44 Finally, having written everything all together and you know exactly what you're looking for, you can start getting an idea of how to get there.

    04:50 Step by step you can begin to gather information and turn unknowns into known's by finding equations for them as you solved the equations you've written down so far.

    About the Lecture

    The lecture Equations of Motion in 1 Dimension by Jared Rovny is from the course Translational Motion. It contains the following chapters:

    • Equations of Motion in 1 Dimension
    • Equations of Motion – Solving Problems

    Included Quiz Questions

    1. The time between the motion variables in their initial and final states
    2. Any time
    3. The time it takes to go from position 0 to position x
    4. The time it takes for the velocity to go from 0 to v
    5. The time it takes to move from x₀ to x at constant velocity v₀
    1. x = x₀ + v₀t + at²/2
    2. x = x₀ + v₀t
    3. x = x₀ + at²/2
    4. x = x₀ + v₀t - at²/2
    5. x = x₀ + v₀t + at²
    1. 0.5 m
    2. 1 m
    3. 2 m
    4. 0.25 m
    5. 4 m

    Author of lecture Equations of Motion in 1 Dimension

     Jared Rovny

    Jared Rovny

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