Chapter 7.1, Problem 62E

Rocket science In the $1999$ movie October Sky, Homer Hickum was accused of launching a rocket that started a forest fire. Homer proved his innocence by showing that his rocket could not have flown far enough to reach where the fire started. He used the following reasoning.

a. Using the fact that $a\left(t\right)=-32$ (see example 11 (b)), find $v\left(t\right)$ and $s\left(t\right)$, given $v\left(0\right)=v_0$ and $s\left(0\right)=0$. (The initial velocity was unknown, and the initial height was $0\text{ft}$.)

b. Homer estimated that the rocket was in the air for $14$ seconds. Use $s\left(14\right)=0$ to find $v_0$.

c. If the rocket left the ground at a $45°$ angle, the velocity in the vertical direction, so the distance traveled horizontally would be $v_{0}t$. (The rocket left the ground at a steeper angle, so this would overestimate the distance from the starting to landing point.) Find the distance the rocket would travel horizontally during its 14-second flight.

Example 11 Velocity and Acceleration

Recall that if the function $s\left(t\right)$ gives the position of a particle at time $t$, then its velocity $v\left(t\right)$ and its acceleration $a\left(t\right)$ are given by

$v\left(t\right)=s^{\prime }\left(t\right)\text{and}a\left(t\right)=v^{\prime }\left(t\right)={s^{\prime }}^{\prime }\left(t\right).$

(b) Many experiments have shown that when an object is dropped, its acceleration (ignoring air resistance) is constant. This constant has been found to be approximately $32\text{ft}$ per second every second; that is,

$a\left(t\right)=-32$

The negative sign is used because the object is falling. Suppose an object is thrown down from the top of the $1100\text{-ft-}$ tall Willis Tower (formerly known as the Sears Tower) in Chicago. If the initial velocity of the object is $-20\text{ft}\text{per}\text{second}$, find $s\left(t\right)$, the distance of the object from the ground at time $t$.