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# Interpreting change in speed from velocity-time graph

Worked example thinking about speed from velocity-time graph.

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• 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? •  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.
• 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? •  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.
• 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? • 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? • 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).
• 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? • Description and title of the video (on youtube, not on the site) do not match its actual content! • What is the difference between speeding up and accelerating? • 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? • 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.  