# What is the average speed of an object that is not moving at t=0 and accelerates at a rate of a(t) =5-t^2/4 on t in [2,4]?

Aug 18, 2017

$\text{average speed} = 12.5$ $\text{m/s}$

#### Explanation:

We're asked to find the average speed of a particle over a time interval, given its acceleration as a function of time.

The equation for average speed is

$\text{average speed" = "distance traveled"/"time interval}$

Position is the second integral of acceleration. The total distance traveled is different from the displacement because it is strictly positive. Therefore, to find the distance traveled from $t \in \left[2 \textcolor{w h i t e}{l} \text{s", 4color(white)(l)"s}\right]$, we take the absolute value (so that it is positive) of the definite integral from $2$ to $4$:

$\text{distance traveled" = int_2^4intcolor(white)(l)|5-(t^2)/4|color(white)(l)dtcolor(white)(l)dt = color(red)(ul(25color(white)(l)"m}$

The time interval is $2$ $\text{s}$, so we have

color(blue)("average speed") = (color(red)(25color(white)(l)"m"))/(2color(white)(l)"s") = color(blue)(ulbar(|stackrel(" ")(" "12.5color(white)(l)"m/s"" ")|)