Average velocity Consider the position function s(t)=sinπt...

Average velocity Consider the position function $s(t)=\sin \pi t$ representing the position of an object moving along a line on the end of a spring. Sketch a graph of $s$ together with a secant line passing through $(0, s(0))$ and $(0.5, s(0.5)) .$ Determine the slope of the secant line and explain its relationship to the moving object.

Key Concepts

Slope of the Secant Line

The slope of the secant line calculated from a position function indicates the average velocity during the time interval between the two points of intersection. It is a measure of how quickly the object's position is changing on average over that period, linking a geometric interpretation with physical motion.

Average Velocity

Average velocity is defined as the total displacement divided by the total time over a given interval. It is graphically represented as the slope of the secant line on the position-time graph, capturing the overall rate of change of the position without accounting for variations within the interval.

Position Function

A position function is a mathematical representation of the location of an object as a function of time. It describes how the position of the object changes over time and is fundamental to understanding motion in one or more dimensions in physics or calculus.

Secant Line

A secant line is a straight line that intersects a curve at two distinct points. In the context of motion, this line connects two points on the position function, showing the average behavior of the function between those points.

Transcript

00:02 In this question we have to first sketch the graph of the object or the position of an object and the function for the position with respect to time is given as s of t as equals to sine of pi t okay so first we have to scale the graph essentially the graph of this will be actually the almost the same graph of um sine x or sine of theta but we first we will have to check for the period so period for this will be equals to 2 pi divided by pi okay so period of this will be actually 5 the maximum value in this case will be 1 and the minimum value will be minus 1 so we can guess the graph now okay so graph will be something like this now so essentially this point is half this point is 1 because the period sorry this is wrong okay because the period is 2 so this point will be 1, this point will be 2, this point will be 3, and this point will be 4.

01:27 Similarly, this point will be minus 1, this point will be minus 2, minus 3 and minus 4.

01:37 This is the maximum point, so the value here will be 1 and the value here we are going to get as minus 1.

01:43 Now in the question we have to discuss the point 0, okay, when t is equal to 0 and when t is equals to 1 or when t is equal to half okay in these two situations you have to discuss so t is equal to half will be this point okay so we'll call these points as well so i'll call origin as o and i'll call this point as p okay so the coordinates of the origin point o will be actually 0 0 comma if i put the value t is equal to 0, so that will be essentially 0.

02:21 And the coordinates of point p, which is when t is equal to half, so the coordinates of that point will be half comma, s of half.

02:33 So if i put s of half here, let me put s of half.

02:40 So the coordinates of that will be half comma, we'll be getting s.

02:44 If you put half here, so it will become pi by two, so sign pi by two is one, so half so we are drawn as second...

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