Q. 48 - Calculus

Question

A bowling ball dropped from a height of $400$ feet will be $s\left(t\right)=400-16t^2$ feet from the ground after $t$ seconds. Use a sequence of average velocities to estimate the instantaneous velocities described below:

     When the bowling ball hits the ground, with $h=-0.5, h=-0.2 and h=-0.1$

Step-by-Step Solution

Answer

Ans:   When $h=-0.5$ instantaneous velocity is: $-152$

          When $h=-0.2$ instantaneous velocity is: $-156.8$

          When $h=-0.1$ instantaneous velocity is:   $-158.4$

Step 1. Given information.

given,  

   A ball is dropped  from a height of $400$ feet and its distance from the ground after $t$ seconds is,  $s\left(t\right)=400-16t^2$

The objective is to estimate the instantaneous velocities for,  

$h=-0.5, h=-0.2 and h=-0.1$

Step 2. First, find the time t when the ball hits the ground.

If the bowling ball hits the ground then $s\left(t\right)=0$

Therefore,

    $\begin{array}{r}400-16t^2=0\\ 16\left(25-t^2\right)=0\\ 25-t^2=0\\ t^2=25\\ t=5\end{array}$

(Take the positive square root)

Step 3. To find the instantaneous velocity for h = - 0 . 5 follows the steps:

     $f\left(t\right)=f\left(5\right)=0$

And

      $f(t+h)=f(4.5)=76$

Therefore the instantaneous velocity is:  

$\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{76-0}{-0.5}\\ =\frac{76}{-0.5}\\ =-152\end{array}$

Step 4. To find the instantaneous velocity for h = - 0 . 2 follows the steps:

$f\left(t\right)=f\left(5\right)=0$

And  $f(t+h)=f(4.8)=31.36$

Therefore the instantaneous velocity is:  

$\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{31.36-0}{-0.2}\\ =\frac{31.36}{-0.2}\\ =-156.8\end{array}$

Step 5. To find the instantaneous velocity for h = - 0 . 1 follows the steps:

$f\left(t\right)=f\left(5\right)=0$

And   $f(t+h)=f(4.9)=15.84$

Therefore the instantaneous velocity is: 

$\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{15.84-0}{-0.1}\\ =\frac{15.84}{-0.1}\\ =-158.4\end{array}$

$f(t+h)=f(4.9)=15.84$