What is the average speed of an object that is still at t=0t=0 and accelerates at a rate of a(t) =2t^2-2t-2a(t)=2t22t2 from t in [2, 3]t[2,3]?

1 Answer
May 26, 2017

-0.5"m"/"s"0.5ms

Explanation:

Since average speed is given by the equation

v_("av-x") = (Deltax)/(Deltat)

We need to find the positions of the car at times t = 2 and t = 3.

We can derive first the velocity equation from the acceleration, and then the position from that velocity equation, via integration, and we'll assume the initial position and velocity are 0:

v_x(t) = int_0^t a_xdt = 2/3t^3 - t^2 -2t

x(t) = int_0^t v_xdt = 1/6t^4 - 1/3t^3 - t^2

Now we have our position equation, x(t), so let's plug in the values 2 and 3 for t to find the position at those times:

x(2) = 1/6(2)^4 - 1/3(2)^3 - (2)^2 = -4.0"m"

x(3) = 1/6(3)^4 - 1/3(3)^3 - (3)^2 = -4.5"m"

So, the average velocity of the car on the interval t in [2,3] is

v_("av-x") = ((-4.5 "m") - (-4.0 "m"))/(3"s" - 2"s") = color(red)(-0.5"m"/"s"