Physics! #15: University Physics 2.4

2.4: Motion with Constant Acceleration

This is a supremely important section! You’re about to learn some equations you will use for a very very long time – the equations for motion with constant acceleration. These are important because there are lots of situations that can be nicely described by constant acceleration, such as dropping a ball or flying through space on a solar sail.

Here, I’m going to list out all the equations. Then I’ll describe how they relate to each ¬†other. And last, I’ll talk about what you’ll do with them.

1) x = x_0 + v_{0x}t + \dfrac{1}{2}a_xt^2

2) v_x = v_{0x} + a_xt

3) {v_x}^2 - {v_{0x}}^2 = 2a_x\Delta x

4) \Delta x = (\dfrac{v_{0x} + v_x}{2})t

Note: I did a slightly different version of equation 3 from the book’s, which I like a little better. Also, whenever the term x - x_0 was used, I just wrote \Delta x. Same thing, but simplifies both the math and the intuition.

Okay, let’s get to it. First, you should notice the relationship between 1 and 2. In fact, any time you see a term that has 1/2 for a coefficient and has a squared variable, you should see the dark footprint of calculus… It means some nice term has been integrated.

The first equation is, in fact, just the integrated version of the second equation. Or, you could say that the second equation is the derivative of the first equation. Same deal, really.

In fact, probably the best way to think of it would be this: You have another equation, let’s call it equation 2-and-a-half, which simply states:

2.5) a_x = c where c is just the rate of acceleration.

You then integrate that and wind up with a formula that looks a lot like equation 2:

\int a_x dt = ct + c_1, where c_1 is just some new constant. You can integrate again to get something that looks a lot like equation 1.

Equations 3 and 4 are just clever rearrangements of the data contained in the other formulas. So, intuitively speaking, the important thing to note here is that, although you’ve got 4 functions, you really have 1 concept: If you want to describe motion in constant acceleration, you need 3 of the following: time, position, velocity, acceleration. The above 4 equations are ways to describe that motion using each possible set of 3.

So, you should definitely memorize these equations. In fact, if you’re being diligent and working all the odd problems (DO IT, SHITHEAD), you’ll memorize them without trying. That said, much like how we talked about all trig functions are variations of sine, all of these motion equations are essentially the same.

Now, let’s talk about equation 3. I think the others are pretty intuitive, but 3 can be a little confusing to think about. Like all the others, it’s easy to use, but you should have a feel for why it makes sense that you can figure out position without knowing how much time has passed.

Here’s how I like to think about it: Although you’re not using the variable t, time is embedded in your equation. Think about a simpler version: say you’re at constant velocity and you travel between two points. By knowing that distance and that velocity, you can figure out how long it took. You can do this because the concept of time passing is a part of velocity. Mathematically, you’re saying “I moved at x meters per y seconds. And I moved x meters. So, I must’ve been moving for y seconds.”

Equation 3 is just another level up. Now you aren’t at constant velocity – you’re at constant acceleration. So, you need to know position change, velocity change, and acceleration. But, the principle is the same.

I think for some reason the idea of deducing passage of time can be a little counter-intuitive. But, of course, we do it all the time. For example, if you fall asleep, you can deduce passage of time by how much light comes through your window.

And those are the equations of motion at constant acceleration. I want to push two things at you: First, you absolutely must work lots of problems. This needs to be working knowledge for you. Second, as always, equations aren’t just tools – they’re ideas. The reason you have 4 equations expressing the same idea is because different tools are needed to find different solutions. HOWEVER, you need to understand why all of this stuff works in real life.


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5 Responses to Physics! #15: University Physics 2.4

  1. Joshua says:

    I think you switched integration and differentiation in your description of how the first two equations relate. Also at one point you have 2_{ax} when you wanted 2a_x it looks like?

    I change equation 3 even more than you do: I divide by 2 (and maybe even multiply by m) to make it clear that this mathematics relates to the conservation of energy.

    And I’m surprised you didn’t mention that equation 4 is saying “distance traveled = average speed * time” where in the case of constant acceleration you can find the average speed by averaging the speed at the start and the speed at the end.

    I wonder if eventually you’ll compile these posts into some short books with titles like “How to Think About Calculus” and “How to Think About Physics”.

  2. david says:

    i can’t see equation 3

  3. Evan says:

    I could use a 4 second video clip of you saying “DO IT, SHITHEAD” as a motivational tactic for my students. Just get a whole collection of modern role models for college students calling them shitheads for academic motivation.

  4. Steuard says:

    Two comments. First, my favorite thing about equation 3 is that it’s secretly just a glimpse of conservation of energy. (Multiply both sides by m/2, apply F=ma, and you’ve got the work-energy theorem in one dimension.) That makes it Awesome(TM).

    And second, as a physics prof, I wish we would all organize our courses so that these equations *didn’t* seem “supremely important”. They’re a very useful special case, to be sure, and they make a lot of instructive examples possible, but a lot of students leave the course thinking that these constant acceleration equations are the heart of physics (because they’re introduced so early and given so much attention). I’d far rather see students feel that way about conservation of energy and momentum, or even Newton’s laws.

    I’d love to see Tom Moore’s “Six Ideas that Shaped Physics” textbooks (or something like them with a “conservation laws first” approach) become more prominent. (Or Chabay and Sherwood’s “Matter and Interactions”, which does some similar things but brings computer simulation into the mix from chapter 1.)

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