# 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]#?

##### 1 Answer

#### Answer:

#### 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

#"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 **absolute value** (so that it is positive) of the definite integral from

#"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

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