Chapter 2.6, Problem 13E

To determine

To find: The velocity of the ball when $t=2$.

Answer to Problem 13E

The velocity of the ball when $t=2$ is $\underset{\_}{-24\text{ ft/s}}$.

Explanation of Solution

Given:

The height (in feet) of the ball after t seconds is $y=40t-16t^2$.

The position function is, $y=40t-16t^2$.

Formula used:

The derivative $f^{\prime }\left(a\right)$ is the velocity $v\left(t\right)$ of the particle at time $t=a$.

$v\left(t\right)=\underset{t\to a}{\lim }\frac{f\left(t\right)-f\left(a\right)}{t-a}$ (1)

Calculation:

Obtain the velocity of the ball when time $t=2$.

Take the position function $y=f\left(t\right)$ and substitute time $a=2$ in equation (1)

$\begin{array}{c}v\left(t\right)=\underset{t\to 2}{\lim }\frac{f\left(t\right)-f\left(2\right)}{t-2}\\ =\underset{t\to 2}{\lim }\frac{\left(40t-16t^2\right)-\left(40\left(2\right)-16{\left(2\right)}^2\right)}{t-2}\\ =\underset{t\to 2}{\lim }\frac{\left(40t-16t^2\right)-\left(80-64\right)}{t-2}\\ =\underset{t\to 2}{\lim }\frac{40t-16t^2-16}{t-2}\end{array}$

$\begin{array}{c}=\underset{t\to 2}{\lim }\frac{-\left(16t^2-40t+16\right)}{t-2}\\ =\underset{t\to 2}{\lim }\frac{-8\left(2t^2-5t+2\right)}{t-2}\end{array}$

Factorize the numerator,

$\begin{array}{c}v\left(t\right)=\underset{t\to 2}{\lim }\frac{-8\left(2t^2-4t-t+2\right)}{\left(t-2\right)}\\ =\underset{t\to 2}{\lim }\frac{-8\left(2t\left(t-2\right)-1\left(t-2\right)\right)}{\left(t-2\right)}\\ =\underset{t\to 2}{\lim }\frac{-8\left(2t\left(t-2\right)-1\left(t-2\right)\right)}{\left(t-2\right)}\\ =\underset{t\to 2}{\lim }\frac{-8\left(2t-1\right)\left(t-2\right)}{\left(t-2\right)}\end{array}$

Cancel the common term $\left(t-2\right)$ from both the numerator and the denominator,

$\begin{array}{c}v\left(t\right)=\underset{t\to 2}{\lim }\left(-8\left(2t-1\right)\right)\\ =-8\underset{t\to 2}{\lim }\left(2t-1\right)\end{array}$

Simplify further and obtain the velocity as follows.

$\begin{array}{c}v\left(t\right)=-8\left(2\left(2\right)-1\right)\\ =-8\left(4-1\right)\\ =-8\left(3\right)\\ =-24\end{array}$

Thus, the velocity of the ball at time $t=2$ is $-24\text{ ft/s}$.