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Road Mirage Angle IOP Physics Education Phys. Educ. 54 P A P ER Phys. Educ. 54 (2019) 065009 (9pp) iopscience.org/ped 2019 Road mirage angle © 2019 IOP Publishing Ltd Michael J Ruiz PHEDA7 Department of Physics, University of North Carolina at Asheville, Asheville, NC 28804, United States of America 065009 E­mail: [email protected] M J Ruiz Abstract Road mirage angle A video is taken of a road­sign mirage from the passenger seat in a car traveling at constant speed on a highway. The video spans the duration of seeing the mirage of the sign, viewing the vanishing of the mirage as the car Printed in the UK approaches, and passing the road sign. The mirage angle, defined as the angle with respect to the horizontal at the moment the observer notes the vanishing of the mirage, can be determined from the video, the speed of the car, and PED known dimensions of the standard road sign. The value can be checked with a theoretical formula using the ambient weather temperament and consulting 10.1088/1361-6552/ab3c28 references to determine the temperature of the air in contact with the road surface. Agreement between observation and theory is within the error bounds. In the conclusion, a quick observational estimate is made in the car 1361­6552 without needing the video. Published Background water gives the desert traveler false hope of get­ When I was 10 years old in 1961 my family made ting a drink. 11 a trip by car from our home in Camden, New Over a body of water the temperature gra­ Jersey, USA to Beaufort, North Carolina, USA. dient can be reversed, a phenomenon called a temper ature inversion. When this occurs, a super­ During the trip, I saw what appeared to be water 6 ior mirage appears above the object. A classic on the road as my dad drove the family car. When case over the English Channel has been described the car approached the water, the water vanished, by Minnaert [1]: observations of magnificent but only to reappear still farther ahead. I asked my ‘ superior mirages were carried out from the dad Why is there water on the road that disap­ “ ” ‘ south coast of England by looking through a tele­ pears? He said, Son, that is a mirage . ’ ‘ ’ scope across the Channel, sometimes in the even­ Mirages are very common and discussed in ing after a very hot day, sometimes while a mist classic texts by Minnaert [1] and Greenler [2]. was lifting’ [1]. An excellent resource on mirages appearing in Superior mirages include towering castles this journal is the featured article by Vollmer [3]. named after Morgan le Fay, who ‘according to The road mirage is classified as an inferior mirage Celtic legend and Arthurian romance, was a fairy, since the image appears below the object, which half­sister of King Arthur, who exhibited her pow­ can be a sign, other cars, or the sky. The cause ers by the mirage’ [4]. This mirage of ‘marvelous is the temperature gradient just above the surface crystal palaces’ is called Fata Morgana, ‘Italian of the hot road. The air becomes hotter and less for Morgan le Fay, or Morgan the fairy’ [4], as it dense as the road surface is approached from was observed over the Strait of Messina in Italy above, resulting in a smaller index of refraction [1]. Ahrens and Henson [5] describe the temper­ near the pavement [1]. Such an inferior mirage ature profile for the Fata Morgana in the follow­ also appears in a hot desert, where the illusion of ing way: ‘The Fata Morgana is observed where 1361­6552/19/065009+9$33.00 1 © 2019 IOP Publishing Ltd M J Ruiz the air temperature increases with height above m sides. The 1.8 m is found from the ratio of the the surface, slowly at first, then more rapidly, then sign’s diagonal to the vertical length of the sup­ slowly again’. port for the sign. The height of the camera in the car from the surface of the road can be measured directly Observing the road mirage with a meter stick when the car is safely parked. In June 2019, I decided to finally revisit Beaufort The result was 1.2 m. The car was traveling 1 1 for the first time since I was a child, this time trav­ 70.0 mi h− = 31.3 m s− and the time from eling with my wife from our home in Asheville, the vanishing· of the mirage· to reaching the sign North Carolina. As we neared Beaufort, driving was 16 s. Therefore the distance from the point on a hot dry highway near New Bern, I saw strik­ where the mirage was last seen to the sign is 1 ing mirages of orange road signs, as shown in fig­ 31.3 m s− 16s = 501 m, shown in figure 3, ure 1. I began shooting a video after it occurred to where the· mirage· angle is α. The lowest point of me that the mirage angle, defined as the observer the light path in figure 3 is not centered since the angle with respect to the horizontal at the moment vertical height of the observer (1.2 m) and height the mirage vanishes, can be determined from a of the sign post (1.8 m) are not equal. Note that video and the constant speed of the car. Knowing the light rays are straight lines except for the very the outside ambient weather temper ature (temper­ short segment close to the road. Before calculat­ ature at the height of the observation) and the ing the mirage angle it is important to discuss this temper ature of the layer of air just above the road feature since often the curvature of the refracted (to be determined), the mirage angle can also be light path is highly exaggerated in many qualita­ calculated from theory. Calculations of the mirage tive treatments of the inferior mirage for illustra­ angle from both observation and theory follow in tion purposes. the next sections. The light path consists of two long straight line segments because the temperature in the air is Mirage angle from observational data constant until the ray is very close to the road sur­ A link to my video, the main source for the obser­ face. Minnaert [1] points out that the air temper­ vational data, is included [6]. The exact moment ature above a hot road drops by 20 °C to 30 °C in the video starts is not important. However, it is the first centimeter above the surface [8]. Students necessary to keep the camera rolling through can verify that air temperature is not perceived to the vanishing of the mirage and until the sign is change until one is almost touching the asphalt, passed by the car. The car should be traveling at where care should be taken because the asphalt a fairly constant speed, which is easy to achieve can cause a burn [9]. Above 1 cm, the lapse­rate is on an uncrowded highway. The left orange sign ‘of the order of 1°–2° C. per cm. at higher levels in the video is shown in figure 2, a detail from a near a heated surface’ [8]. Minnaert describes the screenshot just before passing the sign and about refraction shown in figure 3 as ‘a rather sudden 16 s after the mirage vanished. Each side of the deflection’ [1]. For a pavement temperature of 57 C and standard ‘Low/Soft Shoulder’ diamond­shaped ° sign for a highway in the United States is 3 ft weather temperature of 30 °C [9], a 25 °C drop (0.91 m) [7]. The side of the sign serves as the cal­ in the first cm (average of Minnaert’s 20 °C and ibration to obtain the distance from the bottom of 30 °C) brings the temperature at 1 cm to almost the sign to the ground, a distance found to be 1.8 the ambient weather temperature. An estimate m. The bottom of the sign is important because of the horizontal extent of the curved portion in when the orange of the sign mirage vanishes, the figure 3 on either side of the minimum point is 0.01 m 301 m = 1.7 m, compared to the hundreds last rays reaching the observer come from the 1.80 m · bottom of the sign (least overall refraction of the of metres of the entire horizontal length. Therefore light). To find the 1.8 m, first note that the verti­ the mirage angle can be calculated from the right 1 1.8 cal diagonal of the sign is 0.91 m√2 = 1.3 m, the triangle in figure 3 as α = tan− 301 = 0.34◦. hypotenuse of a right triangle with a pair of 0.91 This angle is very small due to the large horizontal November 2019 2 Phys. Educ. 54 (2019) 065009 Road mirage angle Figure 1. Road mirage which includes two orange signs with a white vehicle between them. Note the familiar mirage appearance of water on this dry hot day. The image is a detail of a frame from the author’s video accompanying this paper [6]. distances of the triangles in figure 3. Next we con­ 0.36°. Therefore, the experimental mirage angle sider the uncertainty. can be reported to two significant figures as First, the road distance of 301 m in figure 3 α = 0.34◦ 0.02◦. (1) is not a direct line from the car to the sign.
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