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+1# on #t in [0,2]#?

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Answer 1 · 2017-04-15T06:57:45

The average speed is #=8.33ms^-1#

The velocity is the integral of the acceleration

#a(t)=2t^2+1#

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

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

Plugging in the initial conditions

#v(0)=3#

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

So,

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

#v(2)=16/3+5=31/3#

The average speed is

#(5-0)barv=int_0^2(2/3t^3+t+3)dt+(31/3*3)#

#5barv=[2/3*1/4*t^4+1/2t^2+3t]_0^2+31#

#=8/3+2+6+31=125/3#

#barv=1/5*125/3=25/3=8.33ms^-1#