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## Main Question or Discussion Point

The following is from my investigation of a problem for my math term paper.

An object is given a certain linear velocity, v(0), has acceleration opposing its velocity, and the object comes to a stop at time = t

d(t

By Fundamental Theorem of Calculus, integral from 0 to t

-> Antiderivative of a(t) = v(t)

-> integral from 0 to t

-> v(t

-> = v(t

So d(t

BUT d = integral from 0 to tf of v(t)dt = integral from 0 to t

So what did I do wrong? I was thinking to take the absolute value of the integral of acceleration, but I don't see how that would make any sense.

Help PLEASE! Any thoughts and comments are also appreciated.

An object is given a certain linear velocity, v(0), has acceleration opposing its velocity, and the object comes to a stop at time = t

_{f}due to the acceleration.d(t

_{f}) = integral from 0 to tf of v(t)dt = integral from 0 to t_{f}of [v(0) + integral from 0 to t_{f}of a(t)dt]dt, d(t) is distance function of time t, v(t) is velocity function of time, and a(t) is acceleration function of time.By Fundamental Theorem of Calculus, integral from 0 to t

_{f}of a(t)dt = antiderivative of a(t_{f}) -antiderivative of a(0)-> Antiderivative of a(t) = v(t)

-> integral from 0 to t

_{f}of a(t)dt = v(t_{f}) - v(0)-> v(t

_{f}) = 0 since the cube comes to stop-> = v(t

_{f}) - v(0) = -v(0)So d(t

_{f}) = integral from 0 to t_{f}of [v(0) + -v(0)]dt = 0 -> BUT d ≠ 0BUT d = integral from 0 to tf of v(t)dt = integral from 0 to t

_{f}of [v(0) + integral from 0 to t_{f}of a(t)dt]dt is TRUE, and integral from 0 to t_{f}of a(t)dt = v(t_{f}) - v(0) = -v(0) is also TRUE.So what did I do wrong? I was thinking to take the absolute value of the integral of acceleration, but I don't see how that would make any sense.

Help PLEASE! Any thoughts and comments are also appreciated.

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