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]#?
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
#"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"" ")|)#