Chapter 10.4, Problem 51E

a.

To determine

Find the -coordinate of the position of the ship when the time difference between the pulses from the transmitting stations is $1000$ microseconds ( $0.001$ second).

Answer to Problem 51E

  $\boxed{x=110.28}$

Explanation of Solution

Given information:

Long distance radio navigation for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light ( $186,000$ miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations, $300$ miles apart, are positioned on a rectangular coordinate system at points with coordinates $\left(-150,0\right)$ and y $\left(150,0\right)$ and that a ship is traveling on a hyperbolic path with coordinates y $\left(x,75\right)$ (see figure)

  

Find the -coordinate of the position of the ship when the time difference between the pulses from the transmitting stations is 1 $1000$ microseconds (3 $0.001$ second).

Calculation:

Consider coordinates $\left(-150,0\right)$ and $\left(150,0\right)$ the centre of the hyperbola $\left(0,0\right)$ it is given that $1unit=1mile$ the ship is travelling on a hyperbolic path with position of foci is same as transmitting stations. Now drive the equation of a hyperbolic path.

The time difference between the pulses from the transmitting stations is $1000$ microseconds ( $0.001$ second).

Speed of pulse is $186,000$ miles per second.

We can get the difference between the distance of ship from the transmitting stations.

  

The difference $d$ is given by,

  $\begin{array}{l}d=186000\times 0.001\\ =186\end{array}$

The equation of hyperbola,

  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Difference between foci to point on hyperbola is $2a$

  $\begin{array}{l}2a=186\\ \boxed{a=93}\end{array}$

The distance between centre and focus is, coordinates of focus $\left(150,0\right)$ and centre are $\left(0,0\right)$ .

  $\begin{array}{l}c=\sqrt{{\left(150-0\right)}^2+{\left(0,0\right)}^2}\\ =150\\ now\\ b=\sqrt{c^2-a^2}\\ =\sqrt{{150}^2-{93}^2}\\ \boxed{b=27\sqrt{19}}\end{array}$

Now substitute the calculated values in $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

  $\begin{array}{l}\frac{x^2}{{\left(93\right)}^2}-\frac{y^2}{{\left(27\sqrt{19}\right)}^2}=1\\ \frac{x^2}{8649}-\frac{y^2}{13851}=1\end{array}$

Now substitute $y=75$ in above equation, for getting $x$ coordinate,

  $\begin{array}{l}\frac{x^2}{8649}-\frac{{75}^2}{13851}=1\\ \frac{x^2}{8649}=1+0.4061\\ x^2=12161.43\end{array}$

  $\boxed{x=110.28}$

Hence, the $x$ coordinate is, $\boxed{x=110.28}$

b.

To determine

Determine the distance between the ship and station $1$ when the ship reaches the shore.

Answer to Problem 51E

  $\boxed{57miles}$

Explanation of Solution

Given information:

Long distance radio navigation for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light ( $186,000$ miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations, $300$ miles apart, are positioned on a rectangular coordinate system at points with coordinates $\left(-150,0\right)$ and $\left(150,0\right)$ and that a ship is travelling on a hyperbolic path with coordinates $\left(x,75\right)$ (see figure)

  

Determine the distance between the ship and station $1$ when the ship reaches the shore.

Calculation:

When the ship reaches the shore $y$ coordinate of the ship become zero, so put $y=0$ in standard equation,

  $\begin{array}{l}\frac{x^2}{8649}-\frac{0^2}{13851}=1\\ \frac{x^2}{8649}=1\\ x^2=8649\\ x=93\end{array}$

Now apply distance formula for $\left(93,0\right)$ and $\left(150,0\right)$ we get,

  $\begin{array}{l}d=\sqrt{{\left(150-93\right)}^2+{\left(0-0\right)}^2}\\ =57\end{array}$

Hence, the distance between the ship and station $1$ is, $\boxed{57miles}$

c.

To determine

What should be the time difference between the pulses?

Answer to Problem 51E

  $\boxed{0.0029\sec onds}$

Explanation of Solution

Given information:

Long distance radio navigation for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light ( $186,000$ miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations, $300$ miles apart, are positioned on a rectangular coordinate system at points with coordinates $\left(-150,0\right)$ and $\left(150,0\right)$ and that a ship is travelling on a hyperbolic path with coordinates $\left(x,75\right)$ (see figure)

  

The ship wants to enter a bay located between the two stations. The bay is $30$ miles from station $1$ . What should be the time difference between the pulses?

Calculation:

The bay lies between station $1$ and station $2$ and bay is $30$ miles from station $1$ . So the coordinates of bay are $\left(120,0\right)$

The distance of bay from station $2$ is,

  $\begin{array}{l}d=\sqrt{{\left(120-\left(-150\right)\right)}^2-{\left(0-0\right)}^2}\\ d=270\end{array}$

The difference of distances of bay from both stations is,

  $\begin{array}{l}\Delta d=270-30\\ =240\end{array}$

From distance -time formula we find time difference of receiving pulses.

  $\begin{array}{l}\Delta t=\frac{240}{186000}\\ =0.0029\sec onds\\ \end{array}$

Hence, the time difference between the pulses $\boxed{0.0029\sec onds}$

d.

To determine

What is the position of the ship?

Answer to Problem 51E

  $\boxed{\left(104.4,60\right)}$

Explanation of Solution

Given information:

Long distance radio navigation for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light ( $186,000$ miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations, $300$ miles apart, are positioned on a rectangular coordinate system at points with coordinates $\left(-150,0\right)$ and $\left(150,0\right)$ and that a ship is travelling on a hyperbolic path with coordinates $\left(x,75\right)$ (see figure)

  

The ship is $60$ miles offshore when the time difference in part (c) is obtained. What is the position of the ship?

Calculation:

When the ship is $60$ miles offshore $y$ -coordinate be of the ship are $60$ so put $y=60$ in standard equation,

  $\begin{array}{l}\frac{x^2}{8649}-\frac{{60}^2}{13851}=1\\ \frac{x^2}{8649}=1.26\\ x^2=10897.74\\ x=104.4\end{array}$

Hence, the coordinates of ship are, $\boxed{\left(104.4,60\right)}$