# 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) =5-2t# on #t in [0,3]#?

##### 1 Answer

Feb 8, 2016

12.5

#### Explanation:

Integrate the acceleration to get the velocity as a function of time.

#v(t) = v(0) + int_0^t a(tau) d tau#

#= 8 + int_0^t (5-2tau) d tau#

#= 8 + [5tau-tau^2]_0^t#

#= 8 + 5t - t^2#

Integrate the velocity to get the displacement as a function of time.

#x(t) - x(0) = int_0^t v(tau) d tau#

#= int_0^t (8 + 5tau - tau^2) d tau#

#= [8tau + 5/2tau^2 - 1/3tau^3]_0^t#

#= 8t + 5/2t^2 - 1/3t^3#

From this, we can calculate the displacement from

#x(3)-x(0) = 8(3) + 5/2(3)^2 - 1/3(3)^3 = 37.5#

Divide this by the time (3 seconds) to get the average velocity.

#bar(v) = frac{37.5}{3-0} = 12.5#