Hey, guys. You'll need to know how to solve motion problems involving acceleration. So what I'm going to do in this video is introduce you to the 4 equations of motion or sometimes called the kinematics equations that you absolutely need to know. But more importantly, I'm going to show you when and how to use each one of these equations. So this is really important for you to learn, practice, and master because it sets the stage for the rest of physics. So let's check it out.

So guys, remember for constant velocity, when the acceleration is equal to 0, for motion problems, the only equation we could use is v=ΔxΔt. We had another version of this equation that we called the position equation. Really these two things were just 2 versions of the same equation. Well, a lot of times in physics, you'll find that the acceleration is not zero. And in those cases, you're going to need 4 more equations called the equations of motion, sometimes called kinematics equations or uniformly accelerated motion equations. Now uniformly accelerated motion just means that the acceleration must be constant. And that's what you absolutely need to know about these equations. You can only use them when the acceleration is constant. Now a lot of textbooks will show the derivations and the proofs and all that stuff, which you don't really need to know, but you should memorize them. So I have them listed in this table here. So we're going to talk about each one of these equations. So let's start with the first one, which is that
v = v_{0} + at. This equation actually comes from an equation that we're pretty familiar with. It comes from
a, the acceleration average, is equal to
Δv
Δt
. So you can manipulate this equation to look like this in the same way we can manipulate the v=ΔxΔt to look like the position equation. It's the same idea.

So we have the next one here, which is nothing really, nothing special about it. v2 = v_{0}2 + 2aΔx. This third equation here I've written in 2 different ways because you'll commonly see it written in those two different ways. They look the exact same. The only difference is that this Δx over here can sometimes remember, it can be expanded and rewritten as x minus x_{initial}, and they just move this thing to the other side. So again, it means the same exact thing. You'll often see them, sort of written interchangeably like that. So now the last one over here is Δx = v_{0} + v2t. And this one has a little asterisk because in order to use this, I strongly recommend that you look at your you ask your professors because some textbooks and professors might not allow you to use this. But basically what this equation says is this, v_{0} + v is actually just the average of 2 velocities. So what this is really saying here is Δx = v_{avg}⋅t, which we actually already know.

Okay. So that's really it. So what the important thing you need to know about these equations is that all of them have some combination of all of these 5 variables. And you need to know which equations have what variables in order to get the right answer. So let's go through each one of them really quickly. So we have v = v_{0} + at. So we have v, v_{0}, a and t, but Δx is missing. So I'm going to write the little sad face there. Now we have v, v_{0}, a, and Δx. So that means this has v, v_{0}, a and Δx. But it's missing time. And now for these two equations, for number 3, whatever form you're using, this has Δx, it has v_{0}, t, and a. So this has, Δx, v_{0}, t, and a, but it's missing the final velocity. And now finally, we've got Δx = v_{0} + v2t. So that means it has Δxv_{0}vt, but it's missing the acceleration. So notice that there's a pattern here. Every single one of these equations is missing one of the variables, except the only one that I have it all in common is the initial velocity. So guys, what you need to know about these 5 variables here, is that to solve any motion problem, whether it's a car or a rocket or whatever it is, with these equations, you're always going to need 3 out of the 5 variables to solve. So the whole game here with all these problems is figuring out which of these 5 variables you have and which 3 you can figure out in order to pick the right equation to get whatever you're missing. So in order to show you how that works, let's just go through this example together.

We have a racing car that is starting from rest. It accelerates constantly, which means that we're able to use the UAM equations, which is good. We have a 160 meter track and the car crosses after 8 seconds. We're going to figure out what's the acceleration of the car. So all of these problems, we can always follow these list of steps to get the right answer. Basically, we're going to start off by drawing the diagram and listing off our variables. It's a great visual way to figure out what we have and what we need. So we've got the, initial velocity. You're saying it starts from rest. We got this car here. It's going to travel and it's going to cross the finish line and then it's going to be moving, probably with some final velocity over here. So let's list off our variables. So let's start off with Δx, then we've got v_{0}, we've got v, a and t. Now, basically, now that we've drawn and listed the 5 variables, let's just identify what we know and what our target is. We're told that the track is a 160 meters, so that's the length, that's 160. The time is going to be crosses the finish line after 8 seconds, so that's a time, that's t equals 8. And so what are we trying to find? We're trying to find the acceleration. So this is going to be our target variable a. So that's going to be the question mark. So that's what we've got here. But notice how we only end up with 2 out of the 5 variables. Now whenever this happens, there's going to be a clue inside of the problem that's going to give you that third one. So let's look out here. We have a racing car that starts from rest and accelerates constantly. So what you need to know there is from rest means that the initial velocity is equal to 0. It starts off with a v_{0} of 0. So that's that third variable that you need. So our v_{0} is equal to 0 and now we have 3 out of 5 variables. So we're good to go. We can pick an equation. So now the next step here is we have to pick an equation without this ignored variable. So let's take a look at what happened here. I was given 3 variables, whether numbers or words in the problem. I'm asked for one of them. But this final velocity, I have no information about. It's not asked and it's not given. So this is called the ignored variable over here. It's a variable that is not asked for or given. And so whenever you are picking the equation, you have to pick the equation that contains the variable you're looking for. So for instance, I'm going to pick an equation that contains acceleration, but it excludes the ignored variable here. So basically, I'm going to pick an equation that does not have v inside of it. So let's take a look at our list and figure out which equation that is. Well, the first one has v, so it's not going to be that one. The second one says v2, so it's not going to be that one. The third one, has a, but it does not include v, which is good. And just to be, you know, thorough, the 4th equation has v+v_{0} over 2, so that's also bad. So notice how this always will give you just one equation to use. So from this list here, we're going to pick equation number 3. Δx = V_{0} T, + 12 AT2. So now, let's just go ahead and fill out all the variables. So all the numbers. We've got 160. Initial velocity is 0 times 8, so that just goes away. Plus 12, now we have a, now we have 8^{2}. Notice how we're only ending up with one unknown variable. That's the acceleration. That's what we want. Then we just rearrange and solve. So this 12 goes to the other side, the 8^{2} goes to the other side. So I have a 160 times 2 divided by 8^{2} equals the acceleration. If you go ahead and plug that in, what you're going to get is 5 meters per second squared. So that's our answer over here.

Moving on to part b. So actually that was part a. So part b now, we're going to be looking for what the car's velocity is at the finish line. So in other words, we're going to be looking for what the final velocity is. This is actually the ignored variable in part a. So we're looking for the final velocity over here. You can go through the list of steps. We already have the diagram. We have the 5 variables. We actually know what this acceleration is now. Now it's just equal to 5. Target variable is v. So actually there is no ignored variable anymore because we have 4 out of the 5. That's what makes these problems easier, is that as you continue to solve, you figure out more and more of the variables. So really we can start off with any one of the equations and the easiest one is going to be the first one. So, I'm looking for the final velocity, so I need v_{0} + a times t. I know v_{0}. That's just equal to 0. I have the acceleration from the first part and I have time. So I'm just going to plug this stuff in. Really straightforward, I've got v equals 0 plus, and then I've got 5 for my acceleration, and then 8 seconds. So we end up with a final velocity of 40 meters per second.

Alright, guys. So that's it. We're going to get a lot more practice with this. That's it for this one. Let me know if you have any questions.