What is the average speed of an object that is moving at #8 m/s# at #t=0# and accelerates at a rate of #a(t) =t^2-2t+4# on #t in [0,3]#?

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Answer 1 · 2018-04-03T06:29:29

The average speed is #=13.25ms^-1#

The speed is the integral of the acceleration.

#a(t)=t^2-2t+4#

#v(t)=int(t^2-2t+4)dt=1/3t^3-t^2+4t+C#

Plugging in the initial conditions

#v(0)=8#

#v(0)=0-0+0+C=8#

Therefore,

#v(t)=1/3t^3-t^2+4t+8#

The average speed is

#barv=1/(3-0)int_0^3v(t)dt=1/(3-0)int_0^3(1/3t^3-t^2+4t+8)dt#

#=1/3[1/12t^4-1/3t^3+2t^2+8t]_0^3#

#=1/3(81/12-9+18+24)#

#=13.25ms^-1#