Answer:
![\overline{v}_{@\Delta t=0.01s}=-15.22ft/s, \overline{v}_{@\Delta t=0.005s}=-15.11ft/s, \overline{v}_{@\Delta t=0.002s}=-15.044ft/s, \overline{v}_{@\Delta t=0.001s}=-15.022ft/s](https://tex.z-dn.net/?f=%5Coverline%7Bv%7D_%7B%40%5CDelta%20t%3D0.01s%7D%3D-15.22ft%2Fs%2C%20%5Coverline%7Bv%7D_%7B%40%5CDelta%20t%3D0.005s%7D%3D-15.11ft%2Fs%2C%20%5Coverline%7Bv%7D_%7B%40%5CDelta%20t%3D0.002s%7D%3D-15.044ft%2Fs%2C%20%5Coverline%7Bv%7D_%7B%40%5CDelta%20t%3D0.001s%7D%3D-15.022ft%2Fs)
Step-by-step explanation:
Now, in order to solve this problem, we need to use the average velocity formula:
![\overline{v}=\frac{y_{f}-y_{0}}{t_{f}-t_{0}}](https://tex.z-dn.net/?f=%5Coverline%7Bv%7D%3D%5Cfrac%7By_%7Bf%7D-y_%7B0%7D%7D%7Bt_%7Bf%7D-t_%7B0%7D%7D)
From this point on, you have two possibilities, either you find each individual
and input them into the formula, or you find a formula you can use to directly input the change of times. I'll take the second approach.
We know that:
![t_{f}-t_{0}=\Delta t](https://tex.z-dn.net/?f=t_%7Bf%7D-t_%7B0%7D%3D%5CDelta%20t)
and we also know that:
![t_{f}=t_{0}+\Delta t](https://tex.z-dn.net/?f=t_%7Bf%7D%3Dt_%7B0%7D%2B%5CDelta%20t)
in order to find the final position, we can substitute this final time into the function, so we get:
so we can rewrite our formula as:
![\overline{v}=\frac{29(t_{0}+\Delta t)-22(t_{0}+\Delta t)^{2}-y_{0}}{\Delta t}](https://tex.z-dn.net/?f=%5Coverline%7Bv%7D%3D%5Cfrac%7B29%28t_%7B0%7D%2B%5CDelta%20t%29-22%28t_%7B0%7D%2B%5CDelta%20t%29%5E%7B2%7D-y_%7B0%7D%7D%7B%5CDelta%20t%7D)
will always be the same, so we can start by calculating that, we take the provided function ans evaluate it for t=1s, so we get:
![y_{0}=29t-22t^{2}](https://tex.z-dn.net/?f=y_%7B0%7D%3D29t-22t%5E%7B2%7D)
![y_{0}=29(1)-22(1)^{2}](https://tex.z-dn.net/?f=y_%7B0%7D%3D29%281%29-22%281%29%5E%7B2%7D)
![y_{0}=7ft](https://tex.z-dn.net/?f=y_%7B0%7D%3D7ft)
we can substitute it into our average velocity equation:
![\overline{v}=\frac{29(t_{0}+\Delta t)-22(t_{0}+\Delta t)^{2}-7}{\Delta t}](https://tex.z-dn.net/?f=%5Coverline%7Bv%7D%3D%5Cfrac%7B29%28t_%7B0%7D%2B%5CDelta%20t%29-22%28t_%7B0%7D%2B%5CDelta%20t%29%5E%7B2%7D-7%7D%7B%5CDelta%20t%7D)
and we also know that the initil time will always be 1, so we can substitute it as well.
![\overline{v}=\frac{29(1+\Delta t)-22(1+\Delta t)^{2}-7}{\Delta t}](https://tex.z-dn.net/?f=%5Coverline%7Bv%7D%3D%5Cfrac%7B29%281%2B%5CDelta%20t%29-22%281%2B%5CDelta%20t%29%5E%7B2%7D-7%7D%7B%5CDelta%20t%7D)
so we can now simplify our formula by expanding the numerator:
![\overline{v}=\frac{29+29\Delta t-22(1+2\Delta t+\Delta t^{2})-7}{\Delta t}](https://tex.z-dn.net/?f=%5Coverline%7Bv%7D%3D%5Cfrac%7B29%2B29%5CDelta%20t-22%281%2B2%5CDelta%20t%2B%5CDelta%20t%5E%7B2%7D%29-7%7D%7B%5CDelta%20t%7D)
![\overline{v}=\frac{29+29\Delta t-22-44\Delta t-22\Delta t^{2}-7}{\Delta t}](https://tex.z-dn.net/?f=%5Coverline%7Bv%7D%3D%5Cfrac%7B29%2B29%5CDelta%20t-22-44%5CDelta%20t-22%5CDelta%20t%5E%7B2%7D-7%7D%7B%5CDelta%20t%7D)
we can now simplify this to:
![\overline{v}=\frac{-15\Delta t-22\Delta t^{2}}{\Delta t}](https://tex.z-dn.net/?f=%5Coverline%7Bv%7D%3D%5Cfrac%7B-15%5CDelta%20t-22%5CDelta%20t%5E%7B2%7D%7D%7B%5CDelta%20t%7D)
Now we can factor Δt to get:
![\overline{v}=\frac{\Delta t(-15-22\Delta t)}{\Delta t}](https://tex.z-dn.net/?f=%5Coverline%7Bv%7D%3D%5Cfrac%7B%5CDelta%20t%28-15-22%5CDelta%20t%29%7D%7B%5CDelta%20t%7D)
and simplify
![\overline{v}=-15-22\Delta t](https://tex.z-dn.net/?f=%5Coverline%7Bv%7D%3D-15-22%5CDelta%20t)
Which is the equation that will represent the average speed of the ball. So now we can substitute each period into our equation so we get:
![\overline{v}_{@\Delta t=0.01s}=-15-22(0.01)=-15.22ft/s](https://tex.z-dn.net/?f=%5Coverline%7Bv%7D_%7B%40%5CDelta%20t%3D0.01s%7D%3D-15-22%280.01%29%3D-15.22ft%2Fs)
![\overline{v}_{@\Delta t=0.005s}=-15-22(0.005)=-15.11ft/s](https://tex.z-dn.net/?f=%5Coverline%7Bv%7D_%7B%40%5CDelta%20t%3D0.005s%7D%3D-15-22%280.005%29%3D-15.11ft%2Fs)
![\overline{v}_{@\Delta t=0.002s}=-15-22(0.002)=-15.044ft/s](https://tex.z-dn.net/?f=%20%5Coverline%7Bv%7D_%7B%40%5CDelta%20t%3D0.002s%7D%3D-15-22%280.002%29%3D-15.044ft%2Fs)
![\overline{v}_{@\Delta t=0.001s}=-15-22(0.001)=-15.022ft/s](https://tex.z-dn.net/?f=%20%5Coverline%7Bv%7D_%7B%40%5CDelta%20t%3D0.001s%7D%3D-15-22%280.001%29%3D-15.022ft%2Fs)