What is the average speed, on #t in [0,5]#, of an object that is moving at #8 m/s# at #t=0# and accelerates at a rate of #a(t) =5t^2-2t# on #t in [0,3]#?

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Answer 1 · 2017-06-11T06:23:58

The average speed is #=#

The speed is the integral of the acceleration

#a(t)=5t^2-2t#

#v(t)=int(5t^2-2t)dt#

#=5/3t^3-t^2+C#

Plugging in the initial conditions

#v(0)=0-0+C=8#

Therefore,

#v(t)=5/3t^3-t^2+8#

When #t=3#,

#v(3)=45-9+8=44ms^-2#

When #t in [3,5]#, the speed is #=44ms^-1#

The average speed is #=barv#

#5barv=int_0^3(5/3t^3-t^2+8)dt+2*44#

#5barv=[5/12t^4-1/3t^3+8t]_0^3+88#

#5barv=135/4-9+24+88=136.75#

#barv=136.75/5=27.35ms^-1#