Tell It to the Judge. (a) How fast must you be approaching a red traffic light 1l = 675 nm2 for it to appear yellow 1l = 575 nm2? Express your answer in terms of the speed of light. (b) If you used this as a reason not to get a ticket for running a red light, how much of a fine would you get for speeding? Assume that the fine is $1.00 for each kilometer per hour that your speed exceeds the posted limit of 90 km>h

Video transcript

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem, an airplane is moving towards a green signal light, green light has a wavelength of nanometers. A crew member in the airplane views the light to be blue wavelength equals nanometers. What is the speed of the airplane in terms of the speed of light? The patent owner is to be rewarded for inventing the high speed plane, determine the reward in dollars earned by the patent owner. The reward is calculated as $2. for every kilometer per hour above the usual cruising speed of 920 kilometers per hour. Ok. So we have two different goals that we're trying to accomplish here. We're trying to determine the reward money in dollars earned by the patent owner and we're also trying to determine the speed of the airplane in terms of speed of light. Ok. So we're given some multiple choice answers both for part I which is the speed in terms of C, the speed of light. And we're also given answers for I I which is the dollar amount for the patent owner reward. Ok. So let's read them off to see what our final answer pair might be. A is 0.0526 C and 11.0 million. Ok. And B is 0.105 C and 78.8 million. C is 0.0526 C and 142 million D is 0. C and 284 million. E is 0.0526 C and 39.4 million. And F is 0.105 C and 21.9 million. OK. So to solve for I first stop or part, I, I should say so, first off, let us recall the equation for the relativistic Doppler effect of light. And also note that the source of light is moving towards the observer and let's call it equation one. And that states that the frequency of the observed is equal to the square root of the speed of light plus the velocity divided by the speed of light minus the velocity multiplied by the frequency of the source. OK. Which is gonna be denoted as F subscript zero. OK. So our end goal is to find the speed of the airplane or the velocity. So we need to solve for you using equation one. And we need to relate the frequency to the wavelength which are we, which we are provided in the problem. So we need to relate the frequency to the wavelength that is really important. So we need to rearrange equation one to solve for you the velocity using some algebra. And we can start by squaring both sides to get rid of the square root and work towards isolating you. So when we do that, speeding ahead, we should get U to equal the velocity to equal the speed of light multiplied by the frequency observed divided by the frequency of the source squared minus one, divided by the frequency observed divided by the frequency of the source squared plus one. So now we need to substitute in the wavelength into our frequency equation noting that the following relations are true that the frequency observed is equal to the speed of light divided by the wavelength and that the frequency of the source is equal to the speed of light divided by the wavelength of the source. OK. So we can take it even further to the state that the frequency of that's observed divided by the frequency of the source is equal to the speed of light divided by the wavelength observed divided by the speed of light divided by the wavelength of the source. OK. So when we simplify this, we should get that the wavelength of the source is divided by the wavelength observed. OK. So we can plug this value and back into equation two. So let's do that. So the velocity is equal to the speed of light multiplied by the wavelength of the source divided by the wavelength observed squared minus one, divided by the wavelength of the source divided by the wavelength observed squared plus one. So now we can plug in all of our known variables and solve for the velocity. So let's do that. So the velocity is equal to the speed of light multiplied by 550 nanometers for the of for the wavelength of the source divided by the wavelength observed which was 495 nanometers. So 550 nanometers for the wavelength of the source divided by the wavelength of the observed which was 495 nanometers squared minus one divided by 550 nanometers divided by nanometers squared plus one. So when we plug that into a calculator, we should get that the velocity in terms of C in terms of speed of light is 0.105 C. And that is our answer for part I. So now to solve for part I I, we need to use dimensional analysis or unit conversion. So we need to take 3.15 multiplied by 10 to the power of 7 m per second. And we need to multiply it by one kilometer divided by one. So in one kilometer, there is 1000 m multiplied by, there's 3600 seconds in one hour. And note that this 3.15 multiplied by 10 to the power of seven. Let's pause here for a second. So if you're curious where this came from, all this is, is how we get that value is we get zero point, we take our velocity which was 0.105. And we multiply it by the speed of light which the numerical value for that is 3. multiplied by 10 to the power of 8 m per second. And that's how we get the 3.15 multiplied by 10 to the power of seven meters per second. Ok? So now we need to convert meters per second to kilometers per hour. So that's where our unit conversion comes in. So all the units cancel out just leaving us kilometers per hour. So we get, we should get 1.134 multiplied by 10 to the power of eight kilometers per hour. Ok. So now we need to take our kilometers per hour value which was 1.134 multiplied by 10 to the power of eight kilometers per hour. And we need to multiply it by $2. per one kilometer per hour. So when we plug that into a calculator, we should get two 0.84 multiplied by 10 to the power of $8. Which when you calculate that out, we should get 284 multiplied by 10 to the power of six, which is the same as saying $284 million. And that is our answer for part I I ok. So going back up to look at our multiple choice answers, that means our correct answer pair has to be the letter D I is 0. ci I is $284 million. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.