What is the average speed, on #t in [0,5]#, of an object that is moving at #7 m/s# at #t=0# and accelerates at a rate of #a(t) =t^2-2t# on #t in [0,4]#?

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Answer 1 · 2017-12-05T14:45:26

The average speed is #=8.27ms^-1#

The speed is the integral of the acceleration

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

Then,

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

Plugging in the initial conditions at #t=0#

#v(0)=7=0-0+C#

#C=7#

Therefore,

#v(t)=1/3t^3-t^2+7# in the interval #[0,4]#

After that #t>4#, assume that the speed is constant

#v(4)=1/3*4^3-4^2+8=64/3-8=13.33ms^-1#

The average speed on the interval #[0,5]# is

#5barv=int_0^4(1/3t^3-t^2+7)dt+13.33*1#

#=[1/12t^4-1/3t^3+7t]_0^4+13.33#

#=(1/12*4^4-4^3/3+7*4)+13.33#

#=41.33#

The average speed is

#barv=41.33/5=8.27ms^-1#