A body moves on a coordinate line such that it has a position sequals​f(t)equalstsquaredminus4tpl us3on the interval 0less than or equalstless than or equals7​,with s in meters and t in seconds. a. Find the​ body's displacement and average velocity for the given time interval. b. Find the​ body's speed and acceleration at the endpoints of the interval. c.​ When, if​ ever, during the interval does the body change​ direction?

A)  Δf = - 49 m, B)  v (7)  = -56 m / s, a = - 8 m / s², C)  t = 0.866 s

Explanation:

A) In this exercise ask to find the displacement and the average velocity, give the function of the movement

           f (t) = - 4t² +3

and the range of motion 0≤ t ≤ 7

the displacement is

for t = 0

           f (0) = 3

for t = 7 s

           f (7) = - 4 7² +3

           f (7) = -46 m

the total displacement is

           Δf = f (7) - f (0)

           Δf = -46 - 3

            Δf = - 49 m

the average speed is defined as the displacement between the time interval

           v = Df / Dt

           v = -49 / 7

           v = - 7 m / s

B) the speed and acceleration of the end points of the motion

the speed of defined by

          v =

in this case

         v =

         v = -8t

let's calculate

          v (7) = -8 7

          v (7)  = -56 m / s

acceleration is defined by

          a =

          a = - 8 m / s²

acceleration is constant throughout the movement

C) the point where the direction changes.

This point is a point where the position goes from positive to negative, the point f = 0

        0 = -4t² +3

        t = √¾

        t = 0.866 s

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