# Calculating distance when velocity is given

Mathematics Asked by Aruha on July 27, 2020

How to find the distance in $$0 < t < 2$$ when the velocity is given:

$$v(t) = frac{t^3}{10} – frac{t^2}{20} + 1$$

I have tried the following and got the answer as $$2.267$$

begin{align*} d = int_{0}^{2}v(t)mathrm{d}t & Longleftrightarrow d = int_{0}^{2} left(frac{t^{3}}{10} – frac{t^{2}}{20}+1right)mathrm{d}t\\ & Longleftrightarrow d = left(frac{t^{4}}{40} – frac{t^{3}}{60}+tright)Big|_{0}^{2}\\ & Longleftrightarrow d = 2.267 m end{align*}
Is this approach correct? Kindly help.

## One Answer

There is a distinction between the difference of the particle's position between the instants $$t_{2} > t_{1}$$ and the distance traveled by the particle corresponding to the time interval $$Delta t = t_{2} -t_{1}$$. The first is given by the integral of $$v(t)$$ and the second is given by the integral of $$|v(t)|$$. Depending in which case you are interested, you can calculate the corresponding integral in order to obtain the answer.

Answered by user1337 on July 27, 2020

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