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## Differential Calculus

### Course: Differential Calculus > Unit 4

Lesson 2: Straight-line motion- Introduction to one-dimensional motion with calculus
- Interpreting direction of motion from position-time graph
- Interpreting direction of motion from velocity-time graph
- Interpreting change in speed from velocity-time graph
- Interpret motion graphs
- Worked example: Motion problems with derivatives
- Motion problems (differential calc)
- Total distance traveled with derivatives

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# Interpreting change in speed from velocity-time graph

Let's examine a velocity-time graph to understand an object's motion. By distinguishing between positive and negative slopes, we discern whether the object is speeding up, slowing down, or maintaining its speed. It's a practical application of calculus concepts to real-world motion scenarios!

## Want to join the conversation?

- Why wouldn't we take the derivative of velocity (acceleration) to figure out when the object is speeding up/slowing down or neither? Wouldn't this give us the same info?(21 votes)
- Yes we can use the derivative of the velocity (acceleration), but the situation is tricky. Speeding up is not necessarily the same as increasing velocity (for example when velocity is negative); slowing down is not necessarily the same as decreasing velocity (for example when velocity is negative).

The object is speeding up (**absolute value**of velocity is increasing), not when the acceleration is positive but instead when the velocity and acceleration have the same sign.

The object is slowing down (**absolute value**of velocity is decreasing), not when the acceleration is negative but instead when the velocity and acceleration have opposite signs.

This means that the object is speeding up when the product of the velocity and acceleration is positive, and slowing down when the product of the velocity and acceleration is negative.(50 votes)

- I am confused at why the object is neither speeding up or slowing down at 6 seconds. If I get the derivative for acceleration at 6 seconds, it is negative so doesn't that mean acceleration does exist at that point and the object is speeding up in the negative direction?(17 votes)
- It helps to think about the graph of speed over time rather than the graph of velocity over time. Since speed is the absolute value of velocity, you just have to take all the negative parts of the graph in the video and flip it around along the x-axis in your head. So, instead of starting out negative at t = 0, it would start out positive and go down toward the x-axis until it sharply changes direction and goes upward again at t = 2. Between t = 2 and t = 6, the graph would be the same, but then at t = 6, the graph would again sharply turn upward creating a second trough of the sine wave. If you're having a hard time visualizing this, you can type in y = abs(sin(x)) on https://www.desmos.com/calculator

Now as to whether the speed is increasing or decreasing at t = 6. The change in speed at t = 6 would be the derivative of the curve at that point, but since the curve has a sharp point in t = 6, the derivative is undefined. That's because on the left side, the slope is getting more and more negative. Even infinitesimally close to t = 6, the slope is still negative. But then, on the right side, the slope is getting more and more positive as it gets closer to t = 6. At t = 6, the slope is undefined because coming from both sides, you approach a different slope.

I hope that helps.(30 votes)

- I have a question regarding interpreting speed changes from a graph - wrt to reading the graph at least, it seems like we can interpret speed as increasing as it is moving away from the center of the graph (i.e., away from the x axis in this example), and decreasing as it is moving closer to the center. Is this true?

Thanks in advance!(13 votes) - I am trying to wrap my head around speed vs velocity. So from what I've read so far, speed is the average rate of change of a distance over time, and velocity is an instantaneous change of a disposition over time, and as the distance --> 0, disposition also --> 0, and speed --> velocity. Still don't get it... What does negative velocity exactly mean on this graph? That we go backwards?(6 votes)
- Velocity is a vector quantity. If we're moving along a line, positive velocity means we're moving in one direction, and negative velocity means we're moving in the other direction.

Speed is the magnitude of the velocity vector, and hence is always positive. So if two objects are moving toward each other at 40 m/s, then they have the same speed (of 40 m/s) but opposite velocities (+40 and -40 m/s).(8 votes)

- Can anyone explain why in this video the points (6,0) have a negative velocity when the previous video with the exact same graph he regards the velocity as neither?(4 votes)
- While these videos are using the same graph, this graph in this video is being used to determine the
of the velocity, while the previous video was about the**rate of change**. Also, the velocity here is decreasing, not negative (although the acceleration would be negative). Tell me if you need any more help!**direction of motion**(5 votes)

- Description and title of the video (on youtube, not on the site) do not match its actual content!(4 votes)
- What is the difference between speeding up and accelerating?(3 votes)
- Acceleration just means a change in velocity. So if you are speeding up, you are accelerating. If you are slowing down, you are also accelerating. Hope this helps!(4 votes)

- In your previous video, the velosity when reaches zero, you said neither, but now you said the velocity is decreasing ??I understand one of them is for speed, but you mentioned both? according to your video, the velocity at zero should be neither?(4 votes)
- If we are looking at the slope at a point, aren't we talking about acceleration and not velocity since slope at any point is the derivative of the function at that point?(3 votes)
- We are talking about the change in speed, which isn't velocity and isn't necessarily acceleration, either. Because your speed can't be negative whereas your velocity can, looking at the change in speed of this velocity-time graph would be like finding the acceleration of the absolute value version of that graph, where everything is a positive value.

Looking at the slope at a point is indeed looking at the derivative, and the derivative of speed is the change in speed.(2 votes)

- I understand that as velocity decreases below 0, the speed increases. Can someone help me understand how that works? Because if speed increases, why doesn't the velocity increase too?(1 vote)
- Speed is the magnitude of velocity. Velocity comes with direction, it indicates a speed at a direction. If velocity becomes negative, it is that speed in the opposite direction.(4 votes)

## Video transcript

- [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.