Problem 43 Medium Difficulty

A particle moves along a straight line with equation of motion $ s = f(t) $, where $ s $ is measured in meters and $ t $ in seconds. Find the velocity and the speed when $ t = 4 $.

$ f(t) = 80t - 6t^2 $

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03:18

Daniel J.

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Video Transcript

suppose the position of a particle at time T is defined by the function F f T Which is equal to 80 T -60 square. And in this problem we need to find the velocity and speed at time T equals four. Now to do this we know that velocity, this is equal to f prime of T. And speed is just the absolute value of velocity. So that's the absolute value of f prime of T. Now define the Derivative given T is equal to four. We will use the definition of derivative at a point. So in here we have velocity. The this is equal to F. Prime of four. That's a limit. ST approaches four of f of t minus F before this all over t minus four. So from here we get a limit SD approaches four of fft which is a TT minus 60 squared and then Fo four which is just Evaluating F at four. That will be 80 times four -6 times four squared fish all over, t minus four. So expanding the numerator, we have limit S. T approaches for We have 80 T -60 squared -80 times four. That's 3. 20 Plus six times 16. That's 96. This all over t minus four. Simplifying the numerator, we have limits SD approaches four of you have negative six t squared plus a D t minus 224. This all over t minus four. A factoring out a -2 in the numerator we have limit as T approaches four of negative two times three t squared minus we have 40 T plus You have 112. This all over t minus four. Factoring the numerator completely. We have Limit as T approaches four of negative two times we have T -4 times three t -28. This all over t minus four. And then from here we can cancel out the T -4 and we get limit as T approaches four of -2 times three T -28 and so evaluating a T equals four, we have -2 times three times 4 -28. This is just equal to 32 and so this is the velocity of the function at time T which is equal to four seconds. The units must be meters per second now because this is already positive, then the this is also the speed of the particle. Therefore the velocity and speed of the particle is 32 m/s. This is when T is equal to four seconds.