109. Average velocity Find the average velocity of a projectil...

Question

109. Average velocity Find the average velocity of a projectile whose velocity over the interval 0 ≤ t ≤ π is given by

v(t) = 10 * sin(3t).

Accepted Answer

Welcome back, everyone. A car's velocity is given by V of T equals 12 for T for T between 0 and pi divided by 2 inclusive. What is the average velocity over this interval? A 3 divided by pi, B 6 divided by pi, C 12 divided by pi, and D 0. For this problem, let's recall the average value formula. In this case, we want to evaluate the average value of velocity. We first will have to take 1 divided by the difference between the end points of the interval. So we take the upper value, which is pi divided by 2, and subtract the lower value which is 0. So that's 1 divided by pi divided by 2 minus 0. And then we multiply this fraction by the integral from the lower limit of 0, up to the upper limit of pi divided by 2. Of the velocity function V of TDT. Now let's go ahead and simplify. This is 1 divided by pi divided by 2, which is 2 divided by pi. Integral from 0 to pi divided by 2. And we're integrating 12 sign. Of 40DT. We can factor out 12 because it is a constant, so we get 24 divided by pi. Integral from 0 to pi divided by 2. Of sine of 4 CDC. We get 24 divided by pi. Integrating sign, we have to recall that the integral of sine is negative cosine, so we get negative cosine of 4 C. But because this C composite function with a linear term for C, according to the reverse chain rule, we're also dividing by the slope of 4. And that's our integral. We want to evaluate it from 0 to pi divided by 2. Once again we can factor out the constant, which is -1/4, and we get -6 divided by pi. Of cosine 4 T evaluated from 0 to pi divided by 2. We get -6 divided by pi. In par, we can evaluate the difference between cosine of 4 T at T equals pi divided by 2. So that's 4 pi divided by 2, and that's cosine of 2 pi. Minus cosine of 40 at 0, so that's cosine of 0. This is equal to -6 divided by pi. In Cs cosine of 2 pi is equal to 1, and a cosine of 0 is 1. So we have 1 minus 1, which is 0, multiplied by -6 divided by pi, and that's 0. So the correct answer for this problem corresponds to the answer choice D. Thank you for watching.

109. Average velocity Find the average velocity of a projectile whose velocity over the interval 0 ≤ t ≤ π is given byv(t) = 10 * sin(3t).

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109. Average velocity Find the average velocity of a projectile whose velocity over the interval 0 ≤ t ≤ π is given by
v(t) = 10 * sin(3t).