The figure shows the velocity function for motion along a...

The figure shows the velocity function for motion along a line. Assume the motion begins with an initial position of $s(0) = 1$. Determine the following. a. The displacement between $t = 0$ and $t = 10$. b. The distance traveled between $t = 0$ and $t = 10$. c. The position at $t = 10$. d. A piecewise function for $s(t)$.

Transcript

00:01 So for this question, we're told that we're given the graph of the velocity function, and we're told that the motion begins with an initial position of s of zero equals 1.

00:17 So at time equals zero, the position is at 1.

00:21 And part a asked for the displacement.

00:31 All right.

00:32 In other words, the displacement between time equals zero and time equals 10.

00:40 So in other words, where does the particle going to be at t equals 10? right.

00:47 So the idea here is that we need to calculate this area between the curve and the x -axis.

00:55 And remember, area below the x -axis is going to be negative.

00:58 So if i look at this area right here, right, this is simply a triangle with a height of two and a base of two.

01:06 So this area right here is going to be one -half base of two.

01:12 Height of two, which is simply two.

01:15 And that's a positive two.

01:17 If i look at this area, right, i see that this is a trapezoid.

01:23 So the area for a trapezoid is one half, one base.

01:30 So we'll call this base one here.

01:32 So from two to seven, that's a base of five, plus from three to six, that's a base of three times a height of two.

01:45 So 5 plus 3 is 4.

01:47 Sorry, 5 plus 3 is 8 and then times a half is 4 times 2 is 8.

01:52 And again, this area is below the curve, so it's going to be a negative 8.

01:56 Negative 8...

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