Find the average velocity

Question

If a ball is thrown in the air with a velocity $34$ ft/s, its height in feet $t$ seconds later is given by

$$ y = 34 t − 16 t^2 .$$

Find the average velocity for the time period beginning when $t = 2$ and lasting $0.5$ second, $0.1$ second, $0.05$ second, $0.01$ second and estimate the instantaneous velocity when $t = 2$.

I tried to solve by doing the following:

$y=34(2)-16(2)^2$

$y=68-64$

$y=a$

$4/0.5=8$ft/s

but I was told that answer is incorrect.

What did I do wrong?

Answer 1

You are being asked for the average velocity over the time span $2$ to $2.5$ seconds, among others. To do that one, you need $y(2.5)$ and $y(2)$ Then the average velocity in that span is $\frac {y(2.5)-y(2)}{2.5-2}$

Answer 2

There is enough information here to complete the following process:

Given $y(t)=34t-16t^2$, we have:

$$y(2)=34\cdot 2-16\cdot 4=68-64=4$$

$$y(2.5)=34\cdot 2.5-16\cdot 2.5^2=85-100=-15$$

$$y(2.1)=34\cdot 2.1-16\cdot 2.1^2=71.4-70.56=0.84$$

$$y(2.05)=2.46$$ (using calculator)

$$y(2.01)=3.6984$$ (using calculator)

Now, the remaining task is to calculate $-19\over 0.5$, $-3.16\over 0.1$, $-1.54\over 0.05$, and $-0.3016\over 0.01$ and see if these numbers are approaching a particular number. I got each of these fractions by writing $y(2+t)-y(2)\over t$ for each of the values $t\in \{0.5,0.1,0.05,0.01\}$.