Answer:
[tex]\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[/tex]
Step-by-step explanation:
Now, in order to solve this problem, we need to use the average velocity formula:
[tex]\overline{v}=\frac{y_{f}-y_{0}}{t_{f}-t_{0}}[/tex]
From this point on, you have two possibilities, either you find each individual [tex] y_{f}, y_{0}, t_{f}, t_{0}[/tex] 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:
[tex]t_{f}-t_{0}=\Delta t[/tex]
and we also know that:
[tex]t_{f}=t_{0}+\Delta t[/tex]
in order to find the final position, we can substitute this final time into the function, so we get:
[tex]y_{f}=29(t_{0}+\Delta t)-22(t_{0}+\Delta t)^{2}[/tex]
so we can rewrite our formula as:
[tex]\overline{v}=\frac{29(t_{0}+\Delta t)-22(t_{0}+\Delta t)^{2}-y_{0}}{\Delta t}[/tex]
[tex]y_{0}[/tex] 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:
[tex]y_{0}=29t-22t^{2}[/tex]
[tex]y_{0}=29(1)-22(1)^{2}[/tex]
[tex]y_{0}=7ft[/tex]
we can substitute it into our average velocity equation:
[tex]\overline{v}=\frac{29(t_{0}+\Delta t)-22(t_{0}+\Delta t)^{2}-7}{\Delta t}[/tex]
and we also know that the initil time will always be 1, so we can substitute it as well.
[tex]\overline{v}=\frac{29(1+\Delta t)-22(1+\Delta t)^{2}-7}{\Delta t}[/tex]
so we can now simplify our formula by expanding the numerator:
[tex]\overline{v}=\frac{29+29\Delta t-22(1+2\Delta t+\Delta t^{2})-7}{\Delta t}[/tex]
[tex]\overline{v}=\frac{29+29\Delta t-22-44\Delta t-22\Delta t^{2}-7}{\Delta t}[/tex]
we can now simplify this to:
[tex]\overline{v}=\frac{-15\Delta t-22\Delta t^{2}}{\Delta t}[/tex]
Now we can factor Δt to get:
[tex]\overline{v}=\frac{\Delta t(-15-22\Delta t)}{\Delta t}[/tex]
and simplify
[tex]\overline{v}=-15-22\Delta t[/tex]
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:
[tex]\overline{v}_{@\Delta t=0.01s}=-15-22(0.01)=-15.22ft/s[/tex]
[tex]\overline{v}_{@\Delta t=0.005s}=-15-22(0.005)=-15.11ft/s[/tex]
[tex] \overline{v}_{@\Delta t=0.002s}=-15-22(0.002)=-15.044ft/s[/tex]
[tex] \overline{v}_{@\Delta t=0.001s}=-15-22(0.001)=-15.022ft/s[/tex]
