1. (08.01 HC) A yo-yo is moving up and down a string so that...

1. (08.01 HC) A yo-yo is moving up and down a string so that its velocity at time $t$ is given by $v(t) = 4cos(t)$ for time $t \ge 0$. The initial position of the yo-yo at time $t = 0$ is $x = 4$. Part A: Find the average value of $v(t)$ on the interval $[0, \frac{\pi}{2}]$. (10 points) Part B: What is the displacement of the yo-yo from time $t = 0$ to time $t = \pi$? (10 points) Part C: Find the total distance the yo-yo travels from time $t = 0$ to time $t = \pi$. (10 points)

Transcript

00:01 Hi there.

00:02 In this question we are given the values the function which is v of t is equal to 3 cosine t.

00:09 So in part a it is being asked find the average value of vt.

00:13 So for the average value from 0 to pi over 2 we have a formula.

00:18 So the average value which is equal to 1 over pi over 2 minus 0 times the integral from 0 to pi over 2 and the function which is cosine t d t.

00:31 Let's take the integral of cosine t, which is three times sine t, and the interval is 0 pi over 2.

00:38 And also we have 1 over pi over 2 as the factor, as the coefficient.

00:45 So this is 2 over pi and times the integral of this one, which is 3 times sine pi over 2 and minus 3 times sine 0.

00:55 Sine zero is zero pi over two is one so this is equal to three times two pi over two over pi which is six over pi so this is the average value for this interval and in part b what is the displacement so we got the function v of t so we have to get the s of t function which is the integral of v of t and so we have to just find the value from zero to pi so the displacement between 0 to pi should be this function, which is 3 cosine t d t.

01:31 Let's take the integral, which is 3 times sine t, and the intervals 0 and pi, 3 times sine pi minus.

01:41 This is 3 times sine 0, which is equal to, so that one is equal to 0 minus 0, which is 0.

01:48 So the displacement between 0 and pi, which is 0.

01:53 So that means just this particle is moving in one direction and then come back to again the same point...

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