Question

What is the average speed of an object that is still at `t=0` and accelerates at a rate of `a(t) =t+3` from `t in [2, 4]`?

Answer 1

Use the definition of acceleration and know that with respect to time, `u(0)=0` because it is still. Also, you should give units of measurement (e.g. `m/s`). I didn't use any because you didn't give me.

`u_(aver)=14`

Explanation:

Being still at `t=0` means that for `u=f(t)->u(0)=0`
Starting from the acceleration definition:

`a=(du)/dt`

`t+3=(du)/dt`

`(t+3)dt=du`

`int_0^t(t+3)dt=int_0^udu`

`int_0^(t)tdt+int_0^t3dt=int_0^udu`

`[t^2/2]_0^t+3[t]_0^t=[u]_0^u`

`(t^2/2-0^2/2)+3(t-0)=u-0`

`u(t)=t^2/2+3t`

So the average velocity between times 2 and 4 is:

`u_(aver)=(u(2)+u(4))/2`

`u(2)=2^2/2+3*2=8`

`u(4)=4^2/2+3*4=20`

Finally:

`u_(aver)=(8+20)/2`

`u_(aver)=14`