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) =t^2-5# on #t in [0,3]#?

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Answer 1 · 2018-04-25T14:53:08

The average speed is #=3.92ms^-1#

The speed is the integral of the acceleration

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

#v(t)=1/3t^3-5t+C#

Plugging in the initial conditions

#v(0)=8ms^-1#

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

Therefore,

#v(t)=1/3t^3-5t+8#

#v(3)=9-15+8=2#

#v(4)=64/3-20+8=28/3#

The average speed is

#(5-0)barv=int_0^3(1/3t^3-5t+8)dt+v(3)+v(4)#

#5barv=[1/12t^4-5/2t^2+8t]_0^3+2+28/3#

#=27/4-45/2+24+34/3#

#=19.58#

#barv=19.58/5=3.92ms^-1#