What is the average speed of an object that is moving at #8 m/s# at #t=0# and accelerates at a rate of #a(t) =2-t^2# on #t in [0,2]#?

Question

↳Redirected from "How do I change #int_0^1int_0^sqrt(1-x^2)int_sqrt(x^2+y^2)^sqrt(2-x^2-y^2)xydzdydx# to cylindrical or spherical coordinates?"

1941 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-04-08T06:27:33

The average speed is #=9.33ms^-1#

The velocity is the integral of the acceleration.

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

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

Plugging in the initial conditions

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

So,

#C=8#

#v(t)=2t-1/3t^3+8#

We can calculate the mean speed

#(2-0)barv=int_0^2(2t-1/3t^3+8)dt#

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

#=(4-4/3+16)-(0)#

#=56/3#

Therefore,

#barv=1/2*56/3=28/3=9.33ms^-1#