Acceleration | Physics | Khan Academy

I decided to raise my regular household car with a sports car, say Ferrari. Well, clearly, it's no match for me. It has a very high top speed, but what if we both agree, for the sake of this race, to limit our top speed to say 80 miles an hour? Now, do you think I have a fair shot of winning? Well, let's find out.

Three, two, one! Oh no, why did that happen? We have the same top speed, right? Well, let's look at this speedometer and see if that uses a clue. So let's try that again. Do you see what happened? Well, even though we both—you can see the sports car was able to pick up that speed much more quickly than my car. This idea of how quickly you can speed up or how quickly you can pick up speed is the idea of acceleration.

So, although we both have the same top speed, Ferrari, or any sports car for that matter, has a much higher acceleration than regular cars, and that's why it was able to extend that lead. So how do we define acceleration? Well, in physics, in general, acceleration is defined as the rate of change in velocity. In other words, it's a measure of how quickly your velocity is changing.

We'll take a lot of examples to make this clear, but mathematically, this means acceleration equals the change in velocity. The Delta represents the change in V, which is the velocity change in velocity divided by the time taken for the change. And so this will then equal—how do you calculate the change in velocity? Well, change in any quantity is usually calculated as the final value minus the initial value divided by the time taken for that change.

Now, in the case of Ferrari, let's say it took about two seconds to reach the 80 miles an hour, and my car took a lot of time. I don't know, maybe it took about 10 seconds, or maybe even more, but let's say 10 seconds. So let's calculate quickly what will be their acceleration.

So, the acceleration of the Ferrari would be the final velocity of the Ferrari, that is 80 miles an hour, minus the initial—will start from zero—divided by two seconds. That gives us 40 miles per hour per second. And similarly, what would be the acceleration of my car? I'm just going to call it M for my car. Well, even here, the change in velocity is the same; it goes from 0 to 80, but it takes about 10 seconds to do that. And so that gives us about 8 miles per hour per second.

So look, even though both cars have the same change in velocity, Ferrari is able to change that velocity much more quickly, giving it a higher acceleration. And that's what this number says—it says per second Ferrari picks up about 40 miles per hour of velocity, whereas my car per second only picks about 8 miles per hour of velocity.

So now let's see why acceleration was so tricky for me. First and foremost, miles per hour is not the standard unit; the standard unit is meters per second. So if I convert this 40 miles per hour, it turns out to be about 18 meters per second. Don't worry about the conversion; in this video, we'll tackle that separately. But that means the acceleration in standard terms becomes 18 meters per second per second.

Again, what does it say? Now it's saying that every second my Ferrari is picking up a velocity of 18 meters per second. And so if we simplify the unit, it then becomes meters per second squared. And that's not very intuitive, right? What is meters per second squared? You should always remember it is meters per second per second.

And the same is the case over here: 8 miles an hour, if I convert it to meters per second, it gives me about 3.6 meters per second. So that is 3.6; its velocity increases by 3.6 meters per second per second. Or again, I can say it is 3.6 meters per second squared.

So you need to be mindful about the units. It's meters per second squared. But here's another reason why it was so tricky for me. My brain used to keep saying acceleration is the same thing as velocity. It's not; acceleration is how quickly the velocity is changing. It only exists when velocity changes. One way to check for it is by looking at this speedometer.

But if you don't have a speedometer, then we can look at the motion diagram. For example, if I take the snapshot of our Ferrari every half a second, let's look at what it looks like. What do we see? We see that in the first half second it traveled a little bit of distance; in the next half second, it traveled a little bit more, then it traveled even more, and so on. This tells me, "Aha! The Ferrari must be accelerating," because it's traveling more and more distance in every subsequent time interval.

Therefore, its velocity must be increasing; it is accelerating. And this, by the way, is called an oil drop diagram because if you imagine that your car was dropping oil, leaking oil every half a second in this case, then you would get the oil drops on the road looking something like this.

And that brings me to another point about acceleration. Look, velocity is a vector quantity; it has a direction, right? Which means acceleration is also a vector quantity; it too has a direction. And again, my brain would say, "Hey, acceleration must be the same direction in which the car is moving," but that's not always the case.

Here's how to think about the direction: if the acceleration value turns out to be positive, like in our example, then it has the same direction as the velocity. So in our example, the velocity is upwards, and since the acceleration has a positive value, that means in our example acceleration must also be upwards or forward—whatever you want to call that.

Let's see if we got this. Here's another oil drop diagram, and imagine the car is again going forward. Can you pause the video and think about whether this car is accelerating? And if it is, what is the direction of the acceleration? Pause and try.

Okay, let's see here. I see that the distance between subsequent time intervals is decreasing. So in this example, velocity is decreasing. So it is still changing; therefore, this car is also accelerating. Well, you might say, "If it's slowing down, isn't it deceleration?" You're right, but in physics, acceleration is a much more general word.

Whether you are speeding up or slowing down, the general term is acceleration. As long as the velocity is changing, we will say it's accelerating. So yes, this acceleration is indeed a deceleration. Now, what's the direction of this? Well, for that, I will look at the sign of my acceleration.

This time, since the final velocity will be smaller than the initial velocity, you will see this will be negative. So here, the acceleration would be negative; and negative acceleration means it is in the opposite direction of the velocity. So since the velocity over here is in the forward direction, in this case, acceleration must be backward.

So look, right in front of your eyes, you can see that acceleration and velocity need not be in the same direction. So look, in general, if your object is speeding up, if its velocity is increasing, acceleration and velocity are in the same direction; they would have the same sign. If one is positive, velocity is positive, and acceleration is also positive.

But if your object is slowing down, if it's decelerating, that's when acceleration would be in the opposite direction of the velocity—they would have opposite signs. If velocity is positive, acceleration becomes negative. And again, let's just confirm that with an animation. There you go; deceleration makes sense, right?

Let's take a couple of more examples because you can never take too many examples with acceleration. So here's another oil drop diagram. Let's say the distance between the oil drops are exactly the same; velocity is upwards. What is the direction of the acceleration? Again, pause and try.

Well, if the distance between the oil drops is exactly the same, that means this velocity is not changing; its velocity is a constant. So there is no acceleration, which means in this example, acceleration is zero—constant velocity. So you see, this object could be moving at an extremely high velocity, but that doesn't matter; it's not changing, and therefore, there is no acceleration.

Okay, what if our car is just stationary, sitting over there? What would be its acceleration? Well, this time we don't need our oil drop diagram; we can just imagine this: the velocity would be zero, but more importantly, that velocity stays zero; it's not changing. In that case, also, the acceleration would be zero.

Here's a final oil drop diagram. This time we have again the oil drops are equidistant, same distance between the oil drops, but it's going in a curve—the car is going in a curve. What do you say about the acceleration?

All right, if the distance between them is the same, then I know that the speed must be the same, so the acceleration must be zero, right? Well, we have to be careful; the speed is the same, but velocity involves direction, and clearly, you can see that the direction is continuously changing.

Which means, even though the speed is the same, velocity is changing, and therefore there must be an acceleration in this case. So this car, even though it's going at a constant speed, is accelerating because the velocity is changing. I know it's tricky, but it's always about the change in velocity, not change in speed.

And now, if you're wondering, "What is the direction of this acceleration?" Don't worry about it; we'll tackle direct accelerations in curved motions separately. But anyways, to sum it all up: acceleration is a measure of how quickly your velocity—not speed, velocity—changes. It's measured as change in velocity divided by the time taken for that change.

And therefore its units become this weird meters per second squared. But what does it mean? It's basically telling you how many meters per second change in velocity is happening per second. And when things are moving in a straight line, if you're speeding up, if your velocity is increasing, acceleration and velocity are in the same direction. If the velocity is decreasing, if you're decelerating, if you're slowing down, then they will be in the opposite direction. And of course, if the velocity is not changing, your acceleration would be zero.