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^2-3t# on #t in [0,4]#?

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Answer 1 · 2017-11-29T07:45:06

The average speed is #=38.67ms^-1#

The speed is the integral of the acceleration

#a(t)=5t^2-3t#

Then,

#v(t)=int(5t^2-3t)dt=5/3t^3-3/2t^2+C#

Plugging in the initial conditions at #t=0#

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

#C=8#

Therefore,

#v(t)=5/3t^3-3/2t^2+8# in the interval #[0,4]#

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

#v(4)=5/3*4^3-3/2*4^2+8=320/3-28+8=86.67ms^-1#

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

#5barv=int_0^4(5/3t^3-3/2t^2+8)dt+86.67*1#

#=[5/12t^4-1/2t^3+8t]_0^4+86.67#

#=(5/12*4^2-4^3/2+8*4)+86.67#

#=193.3#

The average speed is

#barv=193.3/5=38.67ms^-1#