Explain why or why not Determine whether the following stateme...

Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.

Accepted Answer

Welcome back, everyone. Suppose an object's velocity is negative throughout the interval from 2 to 6 inclusive. Are the displacement and the distance traveled on this interval equal? A says yes and B says no. For this problem we know that our velocity function on the given interval is negative, so VFT is less than 0. Let's recall the definition of displacement delta X. It is the integral of the velocity function with respect to time between two time values. So in this case between 2 and 6. So that would be the integral, which allows us to evaluate displacement. Now, considering distance D, we're going to set up a very similar integral. Let's recall that we're integrating the absolute value of velocity, right? So we're neglecting the sign of the velocity function, and we're using speed, right? So we're taking the absolute value of the velocity function within our integrant. And we're integrating with respect to time between. 2 and 6. Now, using the definition of the absolute value function, we can show that the absolute value of V of T is going to be equal to negative V of T, right? Based on its nature in terms of a piecewise function. So we can show that our distance is the integral from 2 to 6 of negative V of TDT. And what we can also do is simply factor out the negative sign. We get negative integral from 2 to 6 of the of TDT. And this integral is equal to displacement delta X. Therefore, we have shown that distance in this context is equal to negative displacement and therefore they are not equal to each other. So we can conclude that distance is not equal to delta X because actually it is equal to negative delta X. So the answer to this problem is B. No. Thank you for watching.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.