What is the average speed of an object that is not moving at $t=0$ and accelerates at a rate of $a(t) =6t-9$ on $t in [3, 5]$ ?

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Answer 1 · 2016-01-03T00:59:15

Take the differential definition of acceleration, derive a formula connecting speed and time, find the two speeds and estimate the average.

$u_(av)=15$

The definition of acceleration:

$a=(du)/dt$

$a*dt=du$

$int_0^ta(t)dt=int_0^udu$

$int_0^t(6t-9)dt=int_0^udu$

$int_0^t(6t*dt)-int_0^t9dt=int_0^udu$

$6int_0^t(t*dt)-9int_0^tdt=int_0^udu$

$6*[t^2/2]_0^t-9*[t]_0^t=[u]_0^u$

$6*(t^2/2-0^2/2)-9*(t-0)=(u-0)$

$3t^2-9t=u$

$u(t)=3t^2-9t$

So the speed at $t=3$ and $t=5$ :

$u(3)=3*3^2-9*3=0$

$u(5)=30$

The average speed for $t in[3,5]$ :

$u_(av)=(u(3)+u(5))/2$

$u_(av)=(0+30)/2$

$u_(av)=15$

What is the average speed of an object that is not moving at t=0t=0 and accelerates at a rate of a(t) =6t-9a(t) =6t-9 on t in [3, 5]t in [3, 5]?