We want to study the vertical motion of a stone, to do it, we will find the motion equations of the stone.
We will find that the average velocity in the given interval is 224 ft/s.
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First, we need to find the acceleration of the stone.
The only force acting on the stone (ignoring air resistance) will be the gravitational force, then the acceleration of the stone will be equal to the gravitational acceleration.
[tex]a(t) = -32ft/s^2[/tex]
To get the velocity equation we need to integrate over time, such that the constant of integration will be equal to the initial velocity, which we know is equal to 400ft/s
Then we get:
[tex]v(t) = (-32 ft/s^2)*t + 400ft/s[/tex]
To get the position equation we integrate again, this time the constant of integration will be the initial position, 10 ft.
[tex]p(t) = (-16ft/s^2)*t^2 + (400ft/s)*t + 10ft[/tex]
Now the average velocity is defined as the quotient between the displacement and the time it took to do that displacement, then the average velocity will be:
[tex]av = \frac{p(8s) - p(3s)}{8s - 3s} \\\\av = \frac{( (-16ft/s^2)*(8s)^2 + (400ft/s)*8s + 10ft) - ((-16ft/s^2)*(3s)^2 + (400ft/s)*3s + 10ft)}{5s} = 224 ft/s[/tex]
Thus the average velocity in that interval is 224 ft/s
If you want to learn more, you can read:
https://brainly.com/question/862972
