Chapter 7.3, Problem 46PPS

(a)

To determine

Graph the function for a 10,000 kilogram ship for the domain $0\le v\le 300,000,000$ .Use 300 million meters per second for the speed of light .

Explanation of Solution

Given:

Using the function $m=\frac{m_0}{\sqrt{1-\left(\frac{v^2}{c^2}\right)}}$,Einstein’s theory of relativity states that the apparent mass m of a particle depends on its velocity v.An object that is tracelling extremely fast , close to the speed of light c, will appear to have more mass compared to its mass at rest, $m_0$

Calculation:

The graph of the equation $m=\frac{m_0}{\sqrt{1-\left(\frac{v^2}{c^2}\right)}}$ , by substituting c = $300\times {10}^6=3\times {10}^8$ and $0\le v\le 300,000,000$ , $m_0=10,000$ :

  

(b)

To determine

Write the viewing window did you use to view the graph.

Answer to Problem 46PPS

  $\left[0;300,000,000;100,000,000\right]\times [0;300,000;50,000]$

Explanation of Solution

Given:

Using the function $m=\frac{m_0}{\sqrt{1-\left(\frac{v^2}{c^2}\right)}}$,Einstein’s theory of relativity states that the apparent mass m of a particle depends on its velocity v.An object that is tracelling extremely fast , close to the speed of light c, will appear to have more mass compared to its mass at rest, $m_0$

Calculation:

From part (a) : $0\le v\le 300,000,000$

  

The window used is $\left[0;300,000,000;100,000,000\right]\times [0;300,000;50,000]$

(c)

To determine

Determine the apparent mass m of the ship for speeds of 100 million ,200 million, and 299 million meters per second .

Answer to Problem 46PPS

  $\begin{array}{l}{m}_1=10,607kg\\ m_2=13,416kg\\ m_3=122,577kg\end{array}$

Explanation of Solution

Given:

Using the function $m=\frac{m_0}{\sqrt{1-\left(\frac{v^2}{c^2}\right)}}$,Einstein’s theory of relativity states that the apparent mass m of a particle depends on its velocity v.An object that is tracelling extremely fast , close to the speed of light c, will appear to have more mass compared to its mass at rest, $m_0$

Calculation:

Given $m_i=\frac{10,000}{\sqrt{1-\left(\frac{v_i{}^2}{{\left(3\times {10}^8\right)}^2}\right)}}$

Substitute

  $\begin{array}{l}{v}_1=100\times {10}^6={10}^8\\ v_2=200\times {10}^6=2\times {10}^8\\ v_2=299\times {10}^6\end{array}$

  $\begin{array}{l}{m}_1=\frac{10,000}{\sqrt{1-\left(\frac{{\left({10}^8\right)}^2}{{\left(3\times {10}^8\right)}^2}\right)}}=\frac{10,000}{\sqrt{1-\frac{1}{9}}}\approx 10,607\\ m_2=\frac{10,000}{\sqrt{1-\left(\frac{{\left(2\times {10}^8\right)}^2}{{\left(3\times {10}^8\right)}^2}\right)}}=\frac{10,000}{\sqrt{1-\frac{4}{9}}}\approx 13,416\\ m_3=\frac{10,000}{\sqrt{1-\left(\frac{{\left(299\times {10}^6\right)}^2}{{\left(3\times {10}^8\right)}^2}\right)}}\approx 122,577\end{array}$