60–63. Equivalent constant velocity Consider the following vel...

Question

60–63. Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.

v(t)=2 sin t, for 0≤t≤π

v(t)=2 sin t, for 0≤t≤π

Accepted Answer

Welcome back everyone. Given the velocity function V of T equals 4 of 2 T for time between 0 and 5 divided by 2 inclusive, what constant velocity would result in the same total distance traveled over the given time period? A 4 divided by pi, B pi divided by 4, C pi divided by 8, and D 8 divided by pi. For this problem we will first will find the total distance traveled. Let's recall that the total distance traveled can be found by integrating the absolute value of the velocity function. With respect to time between two time values and in this case we're considering the interval from 0 to pi divided by 2. Now what we can do is simply show that our velocity function in this interval is always non-negative because we have sine of 2 T and when C is between 0 and 5 divided by 2, sine does not become negative. So what we can do is simply integrate. 4 sign of 2 TDT. From 0 to 2 divided by 2. We can factor out the constant, which is 4. We can integrate sign. The integral of sinus negative cosines, we get negative for cosine of 2 T. and because sin 2 T is a composite function with a linear term, we're also dividing by the slope of the linear term, which is 2. And we want to evaluate the result from 0 to pi divided by 2. Let's simplify this is -2 cosine of 2 T between 0 and 5 divided by 2. And now let's evaluate. We got -2 and. Cosine of 2 multiplied by pi divided by 2 is cosine of pi. Minus Cosine of 2 multiplied by 0, that's cosine of 0. Now, let's simplify. We get -2. Cosine of pi is equal to -1, we get -1 minus cosine of 0 is 1. And now -2 multiplied by -2 is equal to 4. We have our total distance traveled. We also know that our time. Delta T Interval is equal to pi divided by 2 minus 0, which is pi divided by 2. And now using the distance formula for a constant velocity, D equals V multiplied by delta T, we can show that V is equal to D divided by delta T, right? Our distance is 4, and we're dividing by pi divided by 2. Which is 4 multiplied by 2 divided by pi and that's 8 divided by pi. Therefore, the correct answer for this problem is C. Thank you for watching.

60–63. Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.v(t)=2 sin t, for 0≤t≤π

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60–63. Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.

v(t)=2 sin t, for 0≤t≤π