CONCAVITY: THE SLOPE OF SLOPES

Using the graph below, let's create a table of values:

did you see where I goofed? It should be y divided by x!! Sorry...

The slopes at each point are indicated in the table and are visible on the graph.

**What is the pattern exhibited by the slopes?

**How is that pattern shown on the table?

**How is that pattern illustrated on the graph?

CONCAVITY BY DEFINITION

Concavity is the rate at which the subsequent slopes of a graph are changing. Concavity is classified in three different ways:

APPLYING CONCAVITY TO PARTICLE MOTION

Let's take another look at the position versus time graph from the previous section - only this time let's identify the concavity of each interval:

By looking at the graph, we can determine:

**when the slope is positive, the object is MOVING FORWARD
**when the slope is negative, the object is BACKING UP

**when the graph is concave up, the object is ACCELERATING
**when the graph is concave down, the object is DECELERATING

QUESTIONS TO CONSIDER

At which point did the graph change its concavity? What does this point mean in terms of the object's acceleration?



At 10 seconds was the object accelerating? From 10-15 seconds what was true about the object?

CREATING GRAPHICAL REPRESENTATIONS OF MOTION

The motion of an object is described by the following:

Time Interval	Velocity	Acceleration
0 - 4 sec.	Positive	Negative
5 - 6 sec.	Positive	Positive
6 - 10 sec.	Positive	Zero
10 - 16 sec.	Zero	Zero
16 - 18 sec.	Negative	Positive
18 - 20 sec.	Negative	Zero

What would a graphical picture of the object's motion look like?

Would the table below help???

PARTICLE MOTION	WHAT TO GRAPH
Positive Velocity	Positive Slope
Negative Velocity	Negative Slope
Zero Velocity	Horizontal Line
Positive Acceleration	Concave Up
Negative Acceleration	Concave Down
Zero Acceleration	Constant Slope

What WOULD a graphical picture of the object's motion look like?

The graph below represents one possibility...

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