Chapter 1.7, Problem 10QR

Solution

To solve: The given formula for given variable.

The required value of variable $t$ is $\sqrt{\frac{2}{g}\times \left(H-s\right)}$ .

Given data:

The given formula (for an object under free fall from height H ) to be solved for the variable t is $s=H-\frac{1}{2}g{t}^2$ .

Method/Formula used:

Using four mathematical operations addition, multiplication, division, and subtraction isolate the required variable.

Calculations:

The given formula is $s=H-\frac{1}{2}g{t}^2$ , from which variable t is determined as follows:

  $\begin{array}{c}s=H-\frac{1}{2}g{t}^2\leftarrow \text{add }\frac{1}{2}g{t}^2\text{on}\text{both}\text{sides}\\ s+\frac{1}{2}g{t}^2=H-\frac{1}{2}g{t}^2+\frac{1}{2}g{t}^2\\ s+\frac{1}{2}g{t}^2=H\cancel{-\frac{1}{2}g{t}^2}+\cancel{\frac{1}{2}g{t}^2}\leftarrow \text{cancel}\text{}\frac{1}{2}g{t}^2\text{on}\text{the}\text{right}\\ s+\frac{1}{2}g{t}^2=H\leftarrow \text{subtract }s\text{from}\text{both}\text{sides}\\ \cancel{s}-\cancel{s}+\frac{1}{2}g{t}^2=H-s\leftarrow \text{cancel}s\text{on}\text{the}\text{left}\\ \frac{1}{2}g{t}^2=H-s\leftarrow \text{multiply}\text{both}\text{sides}\text{by}\text{2}\\ \cancel{2}\times \frac{1}{\cancel{2}}g{t}^2=2\times \left(H-s\right)\leftarrow \text{cancel}2\text{on}\text{the}\text{left}\end{array}$

Solve further,

  $\begin{array}{l}g{t}^2=2\times \left(H-s\right)\leftarrow \text{divide}\text{both}\text{sides}\text{by}g\\ \frac{\cancel{g}}{\cancel{g}}{t}^2=\frac{2}{g}\times \left(H-s\right)\leftarrow \text{cancel}g\text{on}\text{the}\text{left}\\ t^2=\frac{2}{g}\times \left(H-s\right)\leftarrow \text{take}\text{square}\text{root}\text{of}\text{both}\text{sides}\\ \sqrt{t^2}=\sqrt{\frac{2}{g}\times \left(H-s\right)}\\ t=\sqrt{\frac{2}{g}\times \left(H-s\right)}\end{array}$

Therefore, the value of variable t is $t=\sqrt{\frac{2}{g}\times \left(H-s\right)}$ .