What is the average speed, on #t in [0,5]#, of an object that is moving at #5 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 · 2017-11-16T07:06:13

The average speed is #=6.07ms^-1#

The speed is the integral of the acceleration

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

Then,

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

Plugging in the initial conditions at #t=0#

#v(0)=5=0-0+C#

#C=5#

Therefore,

#v(t)=t^3/3-t^2+5# in the interval #[0,4]#

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

#v(4)=4^3/3-4^2+5=64/3-16+5=10.33ms^-1#

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

#5barv=int_0^4(t^3/3-t^2+5)dt+10.33*1#

#=[t^4/12-t^3/3+5t]_0^4+10.33#

#=(4^2/12-4^3/3+5*4)+10.33#

#=30.33#

The average speed is

#barv=30.33/5=6.07ms^-1#