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AP Calc: CHA‑3 (EU), CHA‑3.B (LO), CHA‑3.B.1 (EK)

- [Instructor] 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, and first, we need to make sure we're reading this carefully, 'cause they're not asking is the velocity 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 can 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 going to actually wanna answer both so that we can see how sometimes, they move together, velocity and speed, but sometimes, one might be increasing while the other might be decreasing. So, if we look at this
point right over here, where our velocity is
two meters per second, the speed is the absolute
value of the velocity, which would also be two meters per second, and we can see that the slope of the velocity-time graph is positive, and 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 of our
velocity is also decreasing. Right at that moment, our
speed is two 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. So, 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. 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, so, 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 is already negative, and it's becoming more negative, so, the absolute value of velocity, which is two meters per second, that is increasing at that moment in time, so, our speed is actually increasing, so, as you 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, so, you could say velocity 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, I'll say, 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,
but what about speed? Well, the speed here is
two meters per second. Remember, it'd be the absolute
value of the velocity, and 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, 'cause the absolute value is getting closer and closer to zero.

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