(a) Sketch a graph of velocity versus time corresponding to the graph of displacement versus tim

(a) Sketch a graph of velocity versus time corresponding to...

Key Concepts

Interpreting Slopes

Interpreting slopes on a displacement-time graph is fundamental in determining instantaneous velocity. A positive slope indicates forward motion (positive velocity), a zero slope indicates the object is momentarily at rest (zero velocity), and a negative slope indicates backward motion (negative velocity).

Identifying Extremes in Velocity

Determining when the instantaneous velocity reaches its greatest positive value involves locating the point on the displacement-time graph with the steepest positive slope. This process also involves identifying moments when the slope is zero (indicating zero velocity) and when it is negative (indicating negative velocity), which are key to fully understanding the object's dynamic motion.

Velocity-Time Graph

A velocity-time graph is a plot that displays the instantaneous velocity of an object along the vertical axis against time on the horizontal axis. It is derived by calculating the slope of the displacement-time graph and provides insights into periods of acceleration, deceleration, and direction changes.

Instantaneous Velocity

Instantaneous velocity is the derivative of the displacement with respect to time. It is visually represented as the slope of the displacement-time graph at any given point, indicating both the speed and the direction of the object's motion at that exact moment.

Displacement-Time Graph

A displacement-time graph represents how an object's position changes with time. Its shape reveals information about the object's motion, where the steepness of the curve indicates the speed, and the curvature can indicate acceleration or deceleration.

Transcript

00:01 In this question, we are given a position versus time graph, and we are asked to sketch the velocity versus time graph and then to interpret it in a number of ways.

00:15 And so what i've set up here is a pair of axes for the velocity time graph, and then i've used these yellow vertical guiding lines to help make sure that key points in time on the position time graph line up accurately on the velocity time graph that i'm about to draw.

00:39 That one needs to move over just a hair.

00:43 Okay, that's better.

00:45 So how do we tell velocity versus time from a position time graph? we need to look at the slope, and i can write that more neatly, we need to look at the slope of a line segment that is tangent to the position time graph curve, i should say, to x versus t curve at the given points.

01:27 So we're going to imagine a little line segment that shows us the slope.

01:34 And so i'm not going to try to use the straight line tool because i think it's going to over complicate the video, but here at a, you know, i'm going to draw in red really quickly because i know that'll stand out really well.

01:47 So red, we kind of have this fairly steep and negative slope.

01:57 B, we have still negative but less steep.

02:01 That's actually not a very good tangent to b.

02:03 So that's better.

02:09 It kind of touches but not great.

02:12 You get the point though that it's less steep and it's negative.

02:17 And then at c, our slope is flat at zero.

02:20 D, we're reasonably steep and positive.

02:26 E, our tangent is back to flat.

02:32 You know, you can lay a straight line like a ruler, a small ruler or a pen along each of these points.

02:42 F is negative, not too steep.

02:47 G is back to zero.

02:49 H is not very steep and positive.

02:52 I is more steep and positive.

02:54 J is not only steep and positive, but it's the same as i.

02:59 These two are pretty close to being same steepness.

03:02 And you'll notice that i to j looks like a straight diagonal line anyway.

03:06 K is still positive but smaller.

03:10 And then l, we're back to flat.

03:13 So how do we now turn this into velocity time graph? i'm going to plot the points where the slope is zero first because that's when my velocity is zero.

03:22 So at c, my velocity is at zero.

03:26 At e, my velocity is at zero.

03:29 At g, my velocity is at zero.

03:31 And at l, the velocity is back to zero.

03:38 And now i can go through and compare.

03:42 I think a is the most steep negative that we have.

03:49 And so a is going to be a negative value that's going to be the furthest from the horizontal axis, the time axis.

04:05 B is to be somewhat close to c, and it's going to be still negative but smaller...