Question

What is the average speed of an object that is moving at `4 m/s` at `t=0` and accelerates at a rate of `a(t) =2t^2-2t+4` on `t in [0,3]`?

Answer 1

`v_a=11,5 " "m/s`

Explanation:

`v(t)=int (2t^2-2t+4) d t`
`v(t)=2/3t^3-2/2t^2+4t+C" for t=0 v=4 m/s C=4"`
`v(t)=2/3 t^3-t^2+4t+4`
`Delta x=int_0^3 (2/3 t^3-t^2+4t+4)d t`
`Delta x=|2/3*1/4 t^4-1/3 t^3+4/2t^2+4t|_0^3`
`Delta x=(1/6*81-27/3+2*9+4*3)-0`
`Delta x=27/2-9+18+12" "Delta x=27/2+21`
`Delta x=(42+27)/2`
`Delta x=69/2 " "Delta x=34,5 m`
`Delta x":displacement at interval (0,3)"`
`v_a=(Delta x)/(Delta t)`
`v_a=(34,5)/(3-0)`
`v_a=(34,5)/3`
`v_a=11,5 " "m/s`