Interpreting direction of motion from position-time graph | AP Calculus AB | Khan Academy

An object is moving along a line. The following graph gives the object's position relative to its starting point over time. For each point on the graph, is the object moving forward, backward, or neither? So pause this video and try to figure that out.

All right, so we see we have position in meters versus time. So for example, this point right over here tells us that after one second, we are four meters ahead of our starting point. Or for example, this point right over here says that after four seconds, we are almost— it seems almost— four meters behind our starting point.

So let's look at each of these points and think about whether we're moving forward, backward, or neither. So at this point right over here, at that moment, we're about two and a half meters in front of our starting point. We're at a positive position of two and a half meters.

But as time goes on, we are moving backwards, closer and closer to the starting point. So this is— we are moving backward. One way to think about it, at this time, we're two and a half meters. If you go forward about half a second, we are then back at our starting point, so we have to go backwards.

And if we look at this point right over here, it looks like we were going backwards this entire time, while our curve is downward sloping. But at this point right over here, when we are about— it looks like five meters behind our starting point— we start going forward again.

But right at that moment, we are going neither forward nor backwards. It's right at that moment where we just finished going backwards, and we're about to go forward. And one way to think about it is, what would be the slope of the tangent line at that point? The slope of the tangent line at that point would be horizontal; and so this is neither.

So we can use that same technique to think about this point. The slope is positive, and we see that. All right, right at that moment, it looks like we are at the starting point. But if you fast forward even a few— even a fraction of a second— we are now in front of our starting point. So we are moving forward.

We are moving forward right over here, and at this point we are at our starting point. But if we think about what's going to happen a moment later, a moment later, we're going to be a little bit behind our starting point. And so here we are moving backwards.

And we're done.