Interpreting change in speed from velocity-time graph | Differential Calculus | Khan Academy

An object is moving along a line. The following graph gives the object's velocity over time. For each point on the graph, is the object speeding up, slowing down, or neither? So pause this video and see if you can figure that out.

All right, now let's do it together. First, we just need to make sure we're reading this carefully because they're not asking if the velocity is increasing, decreasing, or neither. They're saying is the object speeding up, slowing down, or neither? So they're talking about speed, which is the magnitude of velocity. You could think of it as the absolute value of velocity, especially when we're thinking about it in one dimension here.

So even though they're not asking about velocity, I'm actually going to answer both so that we can see how sometimes they move together—velocity and speed—but sometimes they might work differently; one might be increasing while the other might be decreasing. If we look at this point right over here, where our velocity is two meters per second, the speed is the absolute value of velocity, which would also be two meters per second.

We can see that the slope of the velocity-time graph is positive. So our velocity is increasing, and the absolute value of our velocity, which is speed, is also increasing. A moment later, our velocity might be 2.1 meters per second, and our speed would also be 2.1 meters per second. That seems intuitive enough.

Now we get the other scenario. If we go to this point right over here, our velocity is still positive, but we see that our velocity-time graph is now downward sloping. So our velocity is decreasing because of that downward slope, and the absolute value of our velocity is also decreasing. Right at that moment, our speed is 2 meters per second, and then a moment later, it might be 1.9 meters per second.

All right, now let's go to this point. This point is really interesting. Here, we see that our velocity—the slope of the tangent line—is still negative, so our velocity is still decreasing. But what about the absolute value of our velocity, which is speed? Well, if you think about it, a moment before this, we were slowing down to get to a zero velocity, and a moment after this, we're going to be speeding up to start having negative velocity.

You might say, "Wait, speeding up for negative velocity?" Remember, speed is the absolute value. So if your velocity goes from zero to negative one meters per second, your speed just went from zero to one meter per second. Therefore, we're slowing down here, and we're speeding up here, but right at this moment, neither is happening. We are neither speeding up nor slowing down.

Now, what about this point here? The slope of our velocity-time graph, or the slope of the tangent line, is still negative. So our velocity is still decreasing. But what about speed? Well, our velocity has already become negative, and it's becoming more negative, so the absolute value of velocity, which is 2 meters per second, is increasing at that moment in time. So our speed is actually increasing.

So notice here, you see a difference.

Now, what about this point? Well, the slope of the tangent line here of our velocity-time graph is zero right at that point. So that means that our velocity is not changing. You could say velocity is not changing, and if speed is the absolute value or the magnitude of velocity, well, that will also be not changing. So we would say speed is neither slowing down nor speeding up.

Last but not least, this point right over here—the slope of the tangent line is positive, so our velocity is increasing. What about speed? Well, the speed here is two meters per second; remember, it would be the absolute value of the velocity.

Here, the absolute value is actually going down if we forward in time a little bit. So our speed is actually decreasing; we are slowing down as our velocity gets closer and closer to zero because the absolute value is getting closer and closer to zero.