Hey, guys. So in this video, we're going to talk about acceleration, which is a variable that's useful not just in motion problems, but for the rest of physics. So it's really important that you learn this stuff very well. And what we're going to see is that acceleration actually has a pretty simple formula. So let's check it out.

So guys, remember when we talked about velocity? Velocity was a measure of how fast your position changed, and the equation was \( v = \frac{\Delta x}{\Delta t} \), your displacement over time. Well, in a very similar way, the acceleration is a measure of how fast your velocity changes. So let me recap. Again, velocity measures how fast your position changes. Acceleration measures how fast your velocity changes. The equations are very similar. The letter we'll use for acceleration is 'a'. Instead of \( \frac{\Delta x}{t} \), it's \( \frac{\Delta v}{\Delta t} \). The units for this are going to be in meters per second squared. So anytime you see those units, we're always going to be talking about acceleration.

So this equation actually tells us that there are 2 ways that something can have acceleration or can accelerate. It's either the object changes the velocity's magnitude. So either is the change in the velocity's magnitude, or a change in the direction. Because remember that the velocity is a vector and it has both magnitude and direction. So if either one of these things changes, that means that the vector changes and that means that there is some acceleration. So let's actually talk about that. The acceleration is a vector because it depends on a vector. What's different about the acceleration is that it's always a vector. And what I mean by that is that there's no scalar version of the acceleration.

So let's recap all the variables that we know so far. When we study displacement, remember displacement had a scalar equivalent that was a distance that was only magnitude, not direction. And then from the displacement, we got the velocity. The velocity was magnitude and direction but it had a scalar equivalent which was speed. That was just the number, no direction. Well, from the velocity, we can actually get the acceleration. And what's different about this is that there is no scalar equivalent for the acceleration. So acceleration is just always a vector and it always has magnitude and direction.

Alright, guys. So that's really all there is to it. It's a simple equation. So let's just get to some examples. So we've got a car moving to the right at 10 meters per second, and then after some time, we're moving to the right at 30. So let's just draw this out. So we know that velocities are vectors with magnitudes and directions. So I'm going to draw this out. My initial velocity is 10 and we're saying that we're going to the right. So at some later time, now we have a different velocity. So basically, this number here is 30 meters per second is higher and we're also moving to the right. So it's a little bit longer and our v is 30 meters per second. And we know that the time that it took to do this, my \( \Delta t \) was 4 seconds. So now we want the magnitude and the direction of the acceleration. So let's just go through our formula. We know that we want \( a \), and we know that \( a = \frac{\Delta v}{\Delta t} \), change in velocity over change in time. Well, whenever we did \( \Delta x \), we kind of used the \( x_\text{final} - x_\text{initial} \). We expanded that term out. It works the same way here. My change in velocity is just my final velocity minus initial velocity over \( \Delta T \). So my final velocity is 30. My initial velocity is 10 and my \( \Delta T \) is 4. So I just get \( \frac{20}{4} \) and I get 5 meters per second squared. So that's my answer, 5 meters per second squared.

So we know that the acceleration equals 5. But what about the direction? Well, notice how we got a positive number here because we plugged in 2 positive numbers. And we chose the right direction to be positive because that was basically the number that was increasing in this direction. So that means that our acceleration vector points this way.

Let's move on to the second one. We're jogging to the right at 6 meters per second. 3 seconds later, we're jogging to the left at 6 meters per second. So let's draw that out. So we've got initial velocity. My \( v_0 \) is going to be 6 in this direction and then at some later time, now we have a velocity and we're jogging it to the left at 6 meters per second. So that means that my final velocity looks like this. It's the same length of the arrow. It's just backwards. So this final velocity is 6. We have to be careful here because we chose this direction to be positive. That's normally what we do in problems, which means that this direction is negative. So we have a velocity that points to the left, which means that it picks up a negative sign. So it's negative 6. So even though problems don't tell you that it's negative, you're going to have to sometimes infer it from the text or figure it out. What's the magnitude and the direction of the acceleration? So we're just going to use the same exact formula. We also know that the time that it takes for this to happen is \( \Delta t \) is equal to 3 seconds. So my acceleration, we're just going to use my \( \Delta v \) over \( \Delta t \). So now, what's the \( v_\text{final} - v_\text{initial} \)? Well, my \( v_\text{final} \) is going to be negative 6. My \( v_\text{initial} \) is going to be positive 6 and my \( \Delta t \) is 3. So we've got \( -6 - 6 \), which is -12 over 3 and I get -4 meters per second squared. So now that's the magnitude, that magnitude is 4. Now, what about the direction? We know that \( a = 4 \). What about the direction? Well, again, the negative sign usually tells us the direction in physics. So because we got something that points or that has a negative sign, then that means that the acceleration vector actually points to the left. So that is the direction.

So I kinda wanted to point out again because I made this point earlier that there are 2 ways that something can have acceleration. Either you change the velocity's magnitude, in this case, we went from 10 to 30, or you can change the direction. In this case, we went from positive 6 to the right, and then negative 6 to the left. So it's the same number. The only thing that changed was the direction, and that's because there's a sign change. So anyway, so that's it for this one. Let me know if you have any questions.