Suppose the position of an object moving horizontally after...

Suppose the position of an object moving horizontally after \(t\) seconds is given by the following function \(s = f(t)\), where \(s\) is measured in feet, with \(s > 0\) corresponding to positions right of the origin. a. Graph the position function. b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left? c. Determine the velocity and acceleration of the object at \(t = 1\). d. Determine the acceleration of the object when its velocity is zero. e. On what intervals is the speed increasing? \(f(t) = 5t^2 - 6t\), \(0 \le t \le 7\) a. Graph the position function. Choose the correct graph below

Transcript

00:01 Here's the solution here given that the velocity versus time graph and here we have to sketch the position versus time graph and acceleration versus time graph corresponding to this graph.

00:12 Okay, so first of all if we are talking about the this graph from zero to two second, there is increasing in velocity.

00:19 And after that, the velocity is decreasing and after that the velocity is also increasing.

00:23 So if we are talking about the increasing velocity with respect to time, that means this is the acceleration.

00:28 And we know that the acceleration is equals to change.

00:30 In velocity divided by time and here the final velocity is plus one because this is the positive one and that one is the negative one so final velocity is positive one minus the initial velocity is minus 1 divided by 2 so here the acceleration is equals to 1 meter per second square so 0 to 2 the velocity is increasing and acceleration is 1 meter per second is square and from 2 second to 4 second the velocity is decreasing so it will be the retardation and the value of retardation is minus 1 meter per second is square so here we can draw it like us that is 0 to 2 second, there will be the positive acceleration and the value is plus 1 meter per second is square.

01:06 And from 2 to 4, there will be a negative acceleration or we can say that there's a retardation.

01:13 And the magnitude of this retardation is minus 1.

01:15 Then again, the velocity is continuously increasing.

01:17 So from 4 second to 6 second, there will be the acceleration with the same magnitude 1 meter per second square.

01:23 And if we are talking about the 6 second to 10 second, the velocity is constant.

01:27 And the velocity is constant.

01:28 That means there will be the acceleration is zero so there will be the zero acceleration after that the velocity is decreasing right so here the acceleration is negative and the value of acceleration is half because if we are talking about the acceleration from 10 second to 12 seconds so here the change in velocity is minus 1 divided by 2 so that's why minus half meter per second is square so the acceleration is half right so that is minus 1 divided so that is minus 1 divided 2 so that's why minus 1 by 2 right so this is the acceleration time graph and so if we are talking about the position time graph so first of all the acceleration is positive but it will move in the negative direction so the position time graph will be like this and after that it will move in the positive direction so it will like this and after six seconds and we know about the six second after six second to 10 second the velocity is constant so here the graph will be straight line and after at 10 seconds if you are talking about the after 10 seconds after 10 second the velocity is negative velocity is decreasing so when the velocity is decreasing there will be a retarded motion and it will like this so this is the position time graph corresponding to velocity time graph now second thing is given that the acceleration time graph and we have to draw the velocity time graph and the position time graph so first of all from 0 to 2 second right this is the 2 that 1 is the 4 6 8 10 and 12 so if you are talking about the 0 to 2 second the acceleration is positive that means the velocity is continuously increasing so here the velocity is continuously increasing and between the 2 to 4 second the acceleration is zero that means the velocity is constant 4 to 6 the acceleration is negative so that's why the velocity is continuously decreasing so that is 0 sorry that is 2 4 6 8 10 and that 1 is 12 so between 4 to 6 second the velocity is continuously decreasing and 6 to 8 second the velocity the acceleration is 0 so that's why here the velocity is constant so the velocity will be constant now if we are talking about between the 8 to 10 seconds the acceleration is positive so velocity is continuously increasing so velocity is continuously increasing and 10 to 12 second the acceleration is negative so the velocity will be decreasing so it is the velocity time graph corresponding to the acceleration time graph now if you are talking about the position time graph so between 2 0 to 2 second the velocity is continuously increasing so that is 2 4 6 8 10 and 12 so the velocity is continuously increasing.

04:01 So here the velocity is continuously increasing, right? so the position is also continuously increasing.

04:06 After that, the velocity is constant...

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