What is the average speed of an object that is moving at #5 m/s# at #t=0# and accelerates at a rate of #a(t) =20-t^2# on #t in [1,5]#?

1 Answer
Apr 4, 2017

The average speed is #=52ms^-1#

Explanation:

The velocity is the integral of the acceleration.

#a(t)=20-t^2#

#v(t)=int(20-t^2)dt#

#=20t-t^3/3+C#

We apply the initial conditions to find #C#

#v(0)=5#

Therefore, #C=5#

So,

#v(t)=20t-t^3/3+5#

We can calculate the average value of the velocity

#(5-1)barv=int_1^5(20t-t^3/3+5)dt#

#=[20/2t^2-t^4/12+5t]_1^5#

#=(250-625/12+25)-(10-1/12+5)#

#=(2675/12)-(179/12)#

#=208#

So,

#barv=208/4=52ms^-1#