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
Nathan L.
Aug 18, 2017

#"average speed" = 12.5# #"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

#"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 [2color(white)(l)"s", 4color(white)(l)"s"]#, we take the absolute value (so that it is positive) of the definite integral from #2# to #4#:

#"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# #"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"" ")|)#

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