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

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Answer 1 · 2017-11-06T06:54:16

The average speed is #=29/3ms^-1#

The speed is the integral of the acceleration

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

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

The initial conditions are

#v=4ms^-1# when #t=0#

Plugging these values in the equation of #v(t)#

#v(0)=2/3*0-1/2*0+C=4#

Therefore,

#C=4#

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

The average speed is

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

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

#4barv=(128/3-8+4)-(0)#

#barv=1/4*116/3=29/3ms^-1#