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Calculus

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A particle moves in a straight line with velocity t^-2 - 1/9 ft/s. Find the total displacement and total distance traveled over the time interval [1,4].

I found out that the total displacement is .4116

But I cannot find the total distance traveled.

  • Calculus -

    Let
    v(t)=velocity function = t^-2 - 1/9
    s(t)=displacement function. = -t/9-1/t
    and
    displacement = s(4)-s(1)= -25/36 - (-10/9) = 5/12 ft.
    If s(t) is monotonically increasing or decreasing, then the displacement equals the distance travelled.

    However, we note that v(3)=0, after which time the velocity reverses in direction.

    So the distance travelled equals
    s(4)-s(3) - [s(3)-s(1)]
    =-25/36 -(-2/3) - [(-2/3)-(-10/9)]
    =-17/36
    Ignore the sign, since distance is a scalar.
    So distance = 17/36 ft.

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