What is the average speed, on #t in [0,5]#, of an object that is moving at #1 m/s# at #t=0# and accelerates at a rate of #a(t) =2t^2-1# on #t in [0,3]#?

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Answer 1 · 2018-01-16T07:07:15

The average speed is #=4ms^-1#

The speed is the integral of the acceleration

#a(t)=2t^2-1#

Then,

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

Plugging in the initial conditions at #t=0#

#v(0)=1=0-0+C#

#C=1#

Therefore,

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

After that #t>3#, assume that the speed is constant

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

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

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

#=[1/6t^4-1/2t^2+t]_0^3+8#

#=(1/6*3^4-1/2*3^2+3)+8#

#=20#

The average speed is

#barv=20/5=4ms^-1#