Motion Along a Line position s=f(t) of an object moving up...

Motion Along a Line position $s=f(t)$ of an object moving up and down on a coordinate line. (a) When is the object moving away from the origin? toward the origin? At approximately what times is the (b) velocity equal to zero? (c) acceleration equal to zero? (d) When is the acceleration positive? negative?

Key Concepts

Position Function

The position function describes the location of an object along a line at any given time. It provides the fundamental spatial information from which other motion characteristics, like velocity and acceleration, are derived.

Velocity Function

The velocity function is the first derivative of the position function with respect to time. It indicates both the speed and the direction in which an object is moving; the sign of the velocity reveals whether the object is moving in one direction or the opposite along the line.

Acceleration Function

The acceleration function is the derivative of the velocity function, or equivalently, the second derivative of the position function. It measures how quickly the velocity is changing and gives insight into speeding up or slowing down, as well as changes in the direction of motion.

Critical Points Analysis

Analyzing when the velocity or acceleration functions equal zero is crucial because these points may indicate moments when the object changes direction, stops momentarily before reversing course, or reaches local extrema in its motion. Such critical points are key to understanding the overall behavior of the motion.

Motion Relative to a Reference Point

Determining whether an object is moving toward or away from a fixed point, such as the origin, involves comparing the direction of motion (given by the sign of the velocity) with the position relative to that point. This analysis connects the abstract mathematical description of motion with real-world movement.

Transcript

00:01 Welcome back, number eight.

00:02 Here we are.

00:04 We're on question 98 of section 4 .4 in thomas ' calculus edition 12.

00:16 So this is section 4 .4, number 98.

00:23 All right, so part a is when is the object moving? wait, what's the first part of the question? when is the object moving away from the origin? when it's moving toward the origin? okay, so i'm not going to recreate the graph.

00:41 Take a look at the graph on page, what page of my own? 213, top left.

00:48 Number 98.

00:49 All right, so it's moving toward the origin.

00:51 Okay, so this is 98a.

00:56 Moving toward the origin.

01:01 So what you got here is a graph with a y values is how far up or how far down the y axis or vertical axis something is moving.

01:11 Okay? and then the horizontal axis is time in seconds.

01:16 So toward the origin, if you just eyeball the graph they give you, i'd say it would be between zero and 1 .5 seconds.

01:30 And then also about about four to 10 seconds.

01:42 And then again, say about from 12, i don't know, let's say 13 .5.

01:47 They say to approximate it, just eyeball it from 12 seconds, for t to about 13 .5 seconds.

01:54 So when is moving away, away from origin? if we take a look at the graph, this 1 ,0 to 1 .5, it's moving up toward the origin, and 4 to 10 it's moving down toward the origin.

02:19 And then 12 to 13 .5 is moving up again toward the origin.

02:23 Okay? so when it's moving away, i'm going to say in between.

02:26 So between 1 .5 seconds and 4 seconds.

02:34 All righty? put a comma here, put a comma here, put a comma here.

02:40 Okay.

02:41 Then between, oh, i'd say 10 and 12 seconds, and then from 13 .5 to 16 seconds.

02:51 This thing only grasps from 0 to 16 seconds.

02:54 That's the domain of our problem.

02:56 So 13 .5 seconds up to 16 seconds.

02:58 So that's the whole domain.

03:02 Okay, now part b, when it's velocity 0, when the slope of the tangent line to this curve is zero.

03:09 So let's say, when is s prime zero? how about that? all right...