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) =5-2t# on #t in [0,4]#?

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Answer 1 · 2018-02-16T07:49:43

The average speed is #=9.53ms^-1#

The speed is the integral of the acceleration.

The acceleration is

#a(t)=5-2t#

The speed is

#v(t)=inta(t)dt=int(5-2t)dt#

#=5t-t^2+C#

Plugging in the initial conditions

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

Therefore,

#v(t)=5t-t^2+5#

The speed when #t=4# is

#v(4)=20-16+5=9ms^-1#

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

#barv*(5-0)=int_0^4(5t-t^2+5)dt+9*1#

#5barv=[5/2t^2-1/3t^3+5t]_0^4+9#

#5barv=(40-64/3+20)-(0)+9=116/3+9=47.67#

#barv=47.67/5=9.53ms^-1#