What is the average speed, on #t in [0,5]#, of an object that is moving at #3 m/s# at #t=0# and accelerates at a rate of #a(t) =2t^2-2# on #t in [0,2]#?

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Answer 1 · 2017-05-02T06:47:55

The average speed is #=3.53ms^-1#

The speed is the integral of the acceleration

#v(t)=inta(t)dt#

#=int(2t^2-2)dt#

#=2/3t^3-2t+C#

Plugging in the initial conditions

#v(0)=0-0+C=3#

Therefore,

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

#v(2)=2/3*2^3-2*2+3=13/3#

The average value is

#(5-0)barv=int_0^2(2/3t^3-2t+3)dt+13/3*3#

#5barv=[2/3*1/4*t^4-2/2t^2+3t]_0^2+13#

#=(8/3-4+6)-(0)+13#

#=14/3+13=53/3#

#barv=53/15=3.53ms^-1#