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

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Answer 1 · 2017-07-10T09:18:38

The average speed is #=21.3ms^-1#

The speed is the integral of the acceleration.

#a(t)=3t^2-t+4#

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

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

Plugging in the initial conditions, #t=0#

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

Therefore,

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

The average speed is

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

#4barv=[1/4t^4-1/6t^3+2t^2]_0^4#

#=(64-32/3+32)-(0)#

#=256/3#

#barv=64/3=21.3ms^-1#