# Thread: finding how far something has gone

1. ## finding how far something has gone

ok this problem seems easy but it still is giving me problems.

A particle that moves along a straight line has velocity v(t) = t^2e^-(2t)

meters per second after t seconds. How many meters will it travel during the first t seconds?

2. Originally Posted by mbrez88
ok this problem seems easy but it still is giving me problems.

A particle that moves along a straight line has velocity v(t) = t^2e^-(2t)

meters per second after t seconds. How many meters will it travel during the first t seconds?
Since v(t) > 0 for all t, the particle does not change direction and so the distance travelled after the first t seconds is the same as the displacement x after the first t seconds.

Since $\displaystyle v = \frac{dx}{dt}$ you're expected to calculate $\displaystyle x = \int t^2 e^{-2t} \, dt$. Integration by parts (twice) is one approach that could be used. To get the constant of integration, you can assume x = 0 when t = 0.

3. Originally Posted by mbrez88
ok this problem seems easy but it still is giving me problems.

A particle that moves along a straight line has velocity v(t) = t^2e^-(2t)

meters per second after t seconds. How many meters will it travel during the first t seconds?
$\displaystyle \int_0^t x^2 \cdot e^{-2x} \, dx$

integration by parts ... tabular method should make it easier.

$\displaystyle \left[-\frac{e^{-2x}}{4}\left(2x^2 + 2x + 1\right)\right]_0^t$