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# Analyzing motion problems: position

Finding the appropriate expression to use when looking for the position at a certain time.

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• We can't calculate the definite integral of this function from 0 to 7, could we ? as it's undefined for t less than 1/3
• Yes that cannot be done, that is why they do the definite integral from 2 to 7 in the video.
• the velocity of the particle v(t) = sqrt(3t-1)
so, v(t) =sqrt(3) * sqrt(t - 1/3 ) so depends on the equation the velcoty of the partcle starts at t = 1/3 with velocty = sqrt(3)
the net change of position from t =1/3 to t =2 is equal ((10*sqrt(5))/9)

So the question of how the distance from the starting point to t= 2 is equal to 8?
• The velocity of the particle doesn't follow the model at t<2
• the particle's position is undefined when t is less than 1/3. What would it look like at that time? Would it just not move ?
• We don't know. The particles motion before t=1/3 is beyond the scope of this model, so it can't tell us anything about that time period.
• If the equation of the velocity holds, what possibly happened in the 1st second? Why not use a piece-wise equation instead?
(1 vote)
• Well, you could just integrate the velocity function, which will give you a position function. Using that, you can find the position at t=1 (which would also be its displacement). Note that you have the value of displacement at t=2, so you can find the exact position function rather than one with the +C in it.

A piecewise function isn't needed here as, well, the velocity could be modelled by a single equation. If the velocity showed different trends over different intervals, then we would need to use piecewise functions to model it.
(1 vote)
• Couldn't you just use the rate of change in position from time 0 to time 7 to find the displacement? So basically option C but without adding the time at t=2. I know that this isn't one of the options for answers for this question but if you were given the problem without multiple-choice answers couldn't you solve it this way?
(1 vote)
• The function given to us has t<(1/3) undefined, so we can't find the rate of change from 0 to 7. We do, however, already know the displacement after 2 seconds, so we would use that combined with the rate from 2 to 7 as in the video.
(1 vote)
• Is the +8 similar to the +C term in indefinite integrals?
(1 vote)
• Whenever he says position, do i just find the displacement?
(1 vote)
• yes when he x-displacement he means x-coordinate, y-displacement he means y-coordinate
(1 vote)
• if i integrate the velocity i would get the position function like this
Position = F(t) + c
if i know position at t=2 , couldn't i find C, and then plug any t to solve for position?
(1 vote)
• Correct. If you were told to solve it, that's what you'd do. Here though, Sal focuses more on the setup, which is really the more important part of calculus, as integration is easy enough.
(1 vote)