# What is the average speed, on #t in [0,5]#, of an object that is moving at #7 m/s# at #t=0# and accelerates at a rate of #a(t) =2t^2+2# on #t in [0,2]#?

##### 1 Answer

#### Answer:

#### Explanation:

The average speed of the object is represented by the equation

Since the acceleration is always positive, the total distance traveled is equal to the total displacement, so the average speed is the same as the average velocity, so we can use the equation

We must therefore find the total displacement of the object after

To find the position equation from the acceleration equation, we need to do an *integration*. We need to integrate the acceleration equation *twice* because position is the second integral of acceleration (velocity lies in between, as the first integral). We can use the equation

to find the velocity equation, and then integrate the velocity equation to find the position equation. Knowing that

the integral of

and the integral of

The initial velocity is

(excluding units)

The equation for position of time from velocity is

Its starting position is assumed to be

(excluding units)

Now, all we have to do is find the position at

Finally, let's plug this back into our average velocity equation: