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# Analyzing motion problems: total distance traveled

AP.CALC:
CHA‑4 (EU)
,
CHA‑4.C (LO)
,
CHA‑4.C.1 (EK)

## Video transcript

Aleksey you received the following problem a particle moves in a straight line with velocity V of T is equal to negative T squared plus 8 meters per second where T is time in seconds at t is equal to 2 the particles distance from the starting point was 5 meters what is the total distance the particle has traveled between T equals 2 and T equals 6 seconds which expression should Alexa use to solve the problem so we don't actually have to figure the actual answer out we just have to figure out what is the appropriate expression so like always pause this video and see if you can work through it on your own so now let's tackle this together so the key question is what is the total distance the particle has traveled between T equals 2 and T equals 6 so we just care what happens between those points we don't care that the particles distance from the starting point was 5 meters at T equals 2 so this right over here is actually unnecessary information so the first thing that you might want us to think about is well maybe distance is just the integral of the velocity function we see that multiple times if you want to find the change in a quantity you just say the starting time and the ending time and then you integrate the rate function so wouldn't it just to be that now we have to be very very careful if the question was what is the displacement for the particle between time equals 2 and x equals 6 this would have been the correct answer so this would be displacement displacement from T equals to 2 T is equal to 6 but they're not saying displacement they're saying total distance the particle has traveled so this is the total path length for the particle so one way to think about it this is you would integrate not the velocity function this would if you integrate velocity you get displacement instead you would integrate the speed function now what is speed it is the magnitude of velocity in one dimension it would just be the absolute value of your velocity function and so the absolute value of the velocity function this would give you integrating the speed this would give you the distance distance from t equals to two T is equal to six and let's see we have that choice right over here the displacement one here this is an interesting distractor but that is not going to be the choice this one right over here V prime of six that gives you the acceleration if you're taking the derivative of the velocity function the acceleration at six seconds that's not what we're interested in and this gives you the absolute difference in velocity who ended between times six and time - that's not what we were trying to figure out either
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