Math Exploration 9.3

Recall that

 

\begin{equation} \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \end{equation}

 

We want to solve this for v/c.

 

Step 1:

\begin{equation} \gamma^{2} = \frac{1}{1-\frac{v^2}{c^2}} \end{equation}

 

Step 2:

\begin{equation} 1-\frac{v^2}{c^2} = \frac{1}{\gamma^2} \end{equation}

 

Step 3:

\begin{equation} \frac{v^2}{c^2} = 1- \frac{1}{\gamma^2} \end{equation}

 

Final step:

\begin{equation} \frac{v}{c} = \sqrt{1- \frac{1}{\gamma^2}} \end{equation}

 

Plugging in γ = 1.99, we have:

 

\begin{equation} \frac{v}{c} = \sqrt{1- \frac{1}{(1.99)^2}} \end{equation}

 

So v/c = 0.86.