Lesson Video: Acceleration Science

In this video, we will learn how to determine the accelerations of objects that change the speed at which they move.

13:58

Video Transcript

In this video, we will learn how to determine the acceleration of objects that change the speed at which they move. Acceleration has to do with change in speed. And to learn about this idea, say we have an object here on a track along which the object can move. So depending on its motion, the object will appear at different positions along the track at different times. Let’s say that at zero seconds of time the object is here, and then as time passes, it remains there. So at one second, two seconds, three seconds and so on, the object stays in place. Based on this, we know the object is moving at a constant speed, and that speed is zero. So the object just stays in place. That’s not so interesting.

But now say we have another object whose motion we’re following. At first, this object too is at the left end of the track. But then after one second passes, the object is here. After two seconds, it’s here. After three seconds, its position is here and so on, on down the track. Over each one second interval, the distance this object moves is the same. That is, the distance it travels between zero seconds and one second is the same as the distance it travels between one second and two seconds and so on for each additional second. This object then is moving at a constant speed. But that speed is not zero.

But now, what if we had yet another object whose position over time looked like this. Just like the other objects, it starts out here at zero seconds. But then these are its positions along the track at one, two, and three seconds of time. What can we say about this object’s motion? To look into it, let’s put marks on our track to indicate distance. Let’s let meters on the track look like this. So after one second, our object has traveled one meter; after two seconds, it’s traveled four meters; and after three seconds, nine meters. Knowing this, we can calculate our object’s speed over each one second time interval. Over the first second, our object moves a distance of one meter. So between zero and one second, our object’s average speed is one meter per second.

But then let’s look between one second and two seconds of time. Over this span, our object goes from one meter up to four meters of distance. That’s a difference of three meters in one second. This proves that the speed of our object is changing over time. It’s not a constant speed like it was in our first two examples. Whenever an object experiences a changing speed, that means that that object is accelerating. In fact, we can define acceleration as a change in speed over time.

This expresses in words what we can also write as a mathematical equation. If we use the letter 𝑣 to represent speed, the letter 𝑡 to represent time, the letter 𝑎 to represent acceleration, and then if we symbolize a change using the Greek letter Δ, then we can combine these symbols so that 𝑎 equals Δ𝑣 over Δ𝑡. Mathematically, this means the same thing as these words. Acceleration is a change in speed over time.

Knowing this, let’s look again at those objects from earlier. Recall that this object here wasn’t moving at all. It had a constant speed of zero. This means that for this object, Δ𝑣, the change in speed is zero, and that means so is its acceleration. Now, what about this object? This object did move at a nonzero speed, but remember that that speed was constant. Even though the object is moving, its change in speed is still zero. This object also has an acceleration of zero. For our last object, though, we know that there is a change in its speed over time. We saw that the object’s average speed went from one meter per second to three meters per second.

So let’s solve for this last object’s acceleration. Its change in speed, Δ𝑣, is three meters per second minus one meter per second. And then, what is Δ𝑡, its change in time? At first, we might think that it’s this time here, two seconds, minus this starting time here, zero seconds. But we must be careful. The change in speed from one up to three meters per second actually took place over this time interval from one second to two seconds. That’s because it’s at this moment after one second has passed that we can say that our object’s average speed is one meter per second. This means that Δ𝑡 in our equation is two seconds minus one second. Our fraction then equals three minus one or two meters per second in the numerator and two minus one or one second in the denominator.

Now, notice the units in this expression. There are meters per second divided by seconds or meters per second per second. These units show us that we really are calculating a change in speed over time. In this case, the units of speed are meters per second, and the units of time are seconds. There’s a way we can simplify these units a bit. Let’s multiply the denominator of this fraction by one divided by seconds. And then, since we’re doing that to the denominator, we do the same thing to the numerator. In the denominator, when we multiply seconds by one over seconds, that’s like multiplying a variable 𝑥 by one divided by 𝑥. These two values multiplied together equal seconds over seconds. And that’s just equal to one.

Then in the numerator, we multiply meters by one and seconds by seconds. That equals meters divided by seconds times seconds or meters divided by seconds squared. Because this one in the denominator doesn’t change the fraction, we can leave it out. Our units simplify to meters per second squared. All this means then that this fraction here equals two divided by one meter per second squared or just two meters per second squared. This is the acceleration of our third object over the first two seconds of its motion.

By working this example, we see there’s another way to write this general equation for acceleration. We can put it in terms of an initial and a final speed of our object. We did this when we used three meters per second and one meter per second earlier. Now, just as there’s such a thing as constant speed, a speed that doesn’t vary over time, so there is constant or uniform acceleration. Clearing some space, let’s now see if our third object maintains this acceleration of two meters per second squared over all of its motion.

We remember that distances were marked out on our track like this. We’re now studying this third interval of time from two to three seconds. Over this one second interval, our object moved five meters, nine minus four. Its average speed, then, is five meters per second, over this particular one second interval. We can now calculate the acceleration of our object between a time of two seconds and the time of three seconds. That acceleration will equal our final speed of five meters per second minus the initial speed of three meters per second divided by a time interval of three seconds minus two seconds. Five minus three is two, and three minus two is one. So once again, we calculate an acceleration of two meters per second squared.

We can say then that the same acceleration still applies to our object motion. So our object has a uniform or constant acceleration. Its speed is changing, but it changes the same amount over every additional second of time. That is, it went from one to three meters per second, a difference of two meters per second over one second, and then from three to five meters per second over one second, once again a difference of two meters per second. Knowing all this about acceleration, let’s look now at a few examples.

Fill in the blank. An object is accelerating when its speed is (A) increasing, (B) decreasing, (C) increasing or decreasing.

Let’s say we had an object here whose speed over time is increasing. This means if we looked at the position of the object at one second intervals, the distances between the positions of the object would increase. We can recall that acceleration is defined as a change in speed over a change in time. Therefore, this object whose speed is increasing over time is accelerating. But notice that for an object’s speed to change, it doesn’t necessarily need to increase.

Say we had a second object here and that we knew the positions of this object at zero, one, two, and three seconds. In this instance, the distances between the object positions with time are getting smaller. But still the speed of the object, Δ𝑣, is changing. We can say that the speed of the pink object is decreasing. The particular name given to an object whose speed decreases over time is deceleration. We can think of deceleration as negative acceleration. It’s a type of acceleration. Since an object’s speed will change whether it’s speeding up or slowing down, we can answer that an object is accelerating when its speed is increasing or decreasing.

Let’s look now at another example.

Which of the following is a correct unit for acceleration? (A) Meters per second, (B) meters per second squared, (C) meters per second quantity squared.

To begin figuring this out, let’s remember what acceleration is. Mathematically, acceleration is a change in speed, Δ𝑣, divided by a change in time. So let’s say we had an object which at a time of zero seconds was moving at a speed of four meters per second. And then two seconds later, the object is moving at seven meters per second. We can calculate the object’s acceleration using this relationship. We’ll do this so we can see what the units of acceleration can be. All right, in the numerator of this fraction, we want to put the change in our object’s speed. That will be equal to its final speed, seven meters per second, minus its initial speed of four meters per second.

Next, let’s think about Δ𝑡, the time that has elapsed. This is equal to two seconds minus zero seconds. So this fraction, the acceleration of our object, is equal to three meters per second divided by two seconds. And here, we’re really only paying attention to the units. We have the units of speed, meters per second, divided by the units of time, seconds. The question is, to which one of our three answer options does this correspond? We can see right away that we won’t choose option (A). A unit of meters per second is a speed, not a change in speed over time.

To see how to choose between options (B) and (C), let’s work on this expression a bit. Right now, we have a fraction here and we have a fraction here. We can make it though so that we’re only working with one overall fraction. To do that, we’re going to multiply the top and bottom of this overall fraction by the same value. We can choose this value to be whatever we want.

But if we make a specific choice and choose it to be one over seconds, then when we multiply in the denominator, we get seconds divided by seconds, which simplifies to one. Then in the numerator, we multiply meters by one and seconds by seconds. That gives meters divided by seconds times seconds or meters per second squared. Dividing meters per second squared by one doesn’t change the value at all. So we’re free to remove the one. This leaves us with meters per second squared, option (B).

Note that this is different from option (C). In option (C), we see meters per second quantity squared, which means we square both numerator and denominator. That gives us meters squared per second squared, while a correct unit for acceleration is meters per second squared.

Let’s look now at one last example.

A car starts accelerating uniformly from rest. After accelerating for three seconds, the car has a speed of 18 meters per second. What is the acceleration of the car?

Let’s say that this dot here is our car. And if we start counting time at zero seconds, we know that at that start since the car begins from rest, it has a speed of zero meters per second. But then, three seconds later, we’re told that the car has a speed of 18 meters per second. To begin solving for acceleration, let’s recall the mathematical equation for acceleration. The acceleration 𝑎 of an object equals its final speed, we’ll call it 𝑣 two, minus its initial speed all divided by the amount of time it takes for the object to change speeds from 𝑣 one to 𝑣 two.

In our case, the initial speed of the object 𝑣 one is zero meters per second, 𝑣 two is 18 meters per second, and the change in time over which this change in speed happens is three seconds minus zero seconds or simply three seconds. So then, here’s what the equation for our object’s acceleration looks like: 18 meters per second minus zero meters per second is 18 meters per second. And 18 divided by three is six. So when we simplify our units, we get an answer of six meters per second squared. This is the acceleration of the car.

Let’s finish our lesson now by reviewing a few key points. In this video, we learned that acceleration is a change in speed over time. As an equation, 𝑎 equals Δ𝑣 over Δ𝑡 or 𝑣 two minus 𝑣 one over Δ𝑡. We also learned that acceleration is measured in units of meters per second squared, that is, a speed in meters per second divided by a time in seconds. And lastly, we learned that uniform acceleration means an object’s speed changes by equal amounts over equal time intervals. This is a summary of acceleration.

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