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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