Chapter 1.5, Problem 110E

Falling-Body Problems Suppose an object is dropped from a height h₀ above the ground. Then its height after t seconds is given by h = −16t² + h₀, where h is measured in feet. Use this information to solve the problem.

128. A ball is dropped from the top of a building 96 ft tall.

1. (a) How long will it take to fall half the distance to ground level?
2. (b) How long will it take to fall to ground level?

To determine

The time taken by ball to fall half the distance to ground level if it is dropped from 96 ft above the ground.

Answer to Problem 110E

The time taken by ball to fall half the distance to ground level is $\underset{\_}{1.73\text{s}}$.

Explanation of Solution

Given:

The height of an object dropped from a height $h_0$ above the ground after t seconds is given by $h=-16t^2+h_0$ where h is measured in feet.

Calculation:

It is given that the ball is dropped from 96 ft above the ground.

The height of the ball will be $\frac{96}{2}=48\text{ft}$ when it reaches half the distance to ground level.

Substitute $48$ for $h$ and $96\text{ft}$ for $h_0$ in the equation $h=-16t^2+h_0$.

$\begin{array}{l}48=-16t^2+96\\ -16t^2=-48\\ t^2=\frac{-48}{-16}\\ t^2=3\end{array}$

Simplify further as follows.

$\begin{array}{l}t=\pm \sqrt{3}\\ t=\pm 1.73\end{array}$

Note that, time will always be positive.

Thus, the time taken by ball to fall half the distance to ground level is $\underset{\_}{1.73\text{s}}$.

To determine

The time taken by the ball to fall to the ground level.

Answer to Problem 110E

The time taken by the ball to fall to ground level is $\underset{\_}{2.5\text{s}}$.

Explanation of Solution

It is given that the ball is dropped from 96 ft above the ground.

The height of the ball will be zero when it reaches the ground.

Therefore, substitute $0$ for $h$ and $96\text{ft}$ for $h_0$ in the equation $h=-16t^2+h_0$.

$\begin{array}{l}0=-16t^2+96\\ -16t^2=-96\\ t^2=\frac{-96}{-16}\\ t^2=6\end{array}$

Simplify further as follows.

$\begin{array}{l}t=\pm \sqrt{6}\\ t=\pm 2.45\end{array}$

Note that, time will always be positive.

Thus, the time taken by the ball to fall to ground level is $\underset{\_}{2.45\text{s}}$.