What is the average speed of an object that is not moving at #t=0# and accelerates at a rate of #a(t) =6t-9# on #t in [3, 5]#?

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Answer 1 · 2016-01-03T00:59:15

Take the differential definition of acceleration, derive a formula connecting speed and time, find the two speeds and estimate the average.

#u_(av)=15#

The definition of acceleration:

#a=(du)/dt#

#a*dt=du#

#int_0^ta(t)dt=int_0^udu#

#int_0^t(6t-9)dt=int_0^udu#

#int_0^t(6t*dt)-int_0^t9dt=int_0^udu#

#6int_0^t(t*dt)-9int_0^tdt=int_0^udu#

#6*[t^2/2]_0^t-9*[t]_0^t=[u]_0^u#

#6*(t^2/2-0^2/2)-9*(t-0)=(u-0)#

#3t^2-9t=u#

#u(t)=3t^2-9t#

So the speed at #t=3# and #t=5#:

#u(3)=3*3^2-9*3=0#

#u(5)=30#

The average speed for #t in[3,5]#:

#u_(av)=(u(3)+u(5))/2#

#u_(av)=(0+30)/2#

#u_(av)=15#