What is the average speed, on #t in [0,5]#, of an object that is moving at #1 m/s# at #t=0# and accelerates at a rate of #a(t) =2t^2+1# on #t in [0,2]#?

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Answer 1 · 2017-05-02T16:56:08

The average speed is #=6.33ms^-1#

The speed is the integral of the acceleration

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

#=int(2t^2+1)dt#

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

Plugging in the initial conditions

#v(0)=0+0+C=1#

Therefore,

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

#v(2)=2/3*2^3+2+1=25/3#

The average value is

#(5-0)barv=int_0^2(2/3t^3+t+1)dt+25/3*3#

#5barv=[2/3*1/4*t^4+1/2t^2+t]_0^2+25#

#=(8/3+2+2)-(0)+25#

#=20/3+25=95/3#

#barv=95/15=6.33ms^-1#