Chapter 16, Problem 69A

To determine

The velocity of car.

Answer to Problem 69A

  $v_{car}=4.65\times {10}^7\text{ m/s}$

Explanation of Solution

Given:

A driver passes his car through a red light.

Wavelength of red light is ${\lambda }_R=645\text{ nm}$

Wavelength of green color is ${\lambda }_G=545\text{ nm}$

Figure 1

Formula used:

Doppler Effect is the change in wavelength or frequency of waves due to the motion of an observer relative to the source or due to the motion of a source relative to an observer. For the given problem, the red-shift velocity equation can be used to find the velocity of a car. That is,

  $v_r=c\left(1-\frac{\lambda }{{\lambda }_r}\right)$

  $\left(1\right)$

Where,

$c$ is the velocity of the light, $c=3\times {10}^8\text{ m/s}$

  $v_r$ is the velocity of an observer or source to cause red shift.

$\lambda$ is the wavelength of source, in this case $\lambda ={\lambda }_G$

${\lambda }_r$ is the red shifted wavelength, in this case ${\lambda }_r={\lambda }_R$

Calculation:

Let $v_r$ can be written as $v_{car}$

$\lambda$ is the wavelength of source, in this case $\lambda ={\lambda }_G$

${\lambda }_r$ is the red shifted wavelength, in this case ${\lambda }_r={\lambda }_R$

Hence equation $\left(1\right)$ can be written as,

  $v_{car}=c\left(1-\frac{{\lambda }_G}{{\lambda }_R}\right)$

  $\left(2\right)$

Substituting the numerical values in equation $\left(2\right)$ ,

  $v_{car}=\left(3\times {10}^8\text{ m/s}\right)\left(1-\frac{545\text{ nm}}{645\text{ nm}}\right)=4.65\times {10}^7\text{ m/s}$

Conclusion:

The driver would have to go with a velocity of $4.65\times {10}^7\text{ m/s}$ .