# What is the average speed of an object that is still at #t=0# and accelerates at a rate of #a(t) = 5t+1# from #t in [0, 6]#?

##### 1 Answer

#### Explanation:

To find the *average speed* of the object, we have to know both the total distance traveled (

We need to integrate this acceleration equation in order to find the *position* equation. The integral of

Doing the same for the second quantity (

Therefore, the first integral of the acceleration equation, which is the *velocity* equation, is

because the initial velocity and position are assumed to be

Since this is the *velocity* equation, we need to integrate this again to find the position equation. Following the same procedure as before, our result is

Now that we have our position equation, let's find the position of the particle at the time

Since the velocity and position are *always* increasing here, the average speed is the same as the average velocity, which is given by the equation

Since our initial position and time are both