A mirror on the passenger side of your car is convex and has...

A mirror on the passenger side of your car is convex and has a radius of curvature with magnitude $18.0 \mathrm{~cm}$. (a) Another car is seen in this side mirror and is $13.0 \mathrm{~m}$ behind the mirror. If this car is $1.5 \mathrm{~m}$ tall, what is the height of its image? (b) The mirror has a warning attached that objects viewed in it are closer than they appear. Why is this so?

Key Concepts

Radius of Curvature and Focal Length Relationship

The radius of curvature (R) of a spherical mirror is directly related to its focal length (f), typically by the relation f = R/2. This relationship is significant because it allows one to determine the focal length given the radius of curvature, which in turn is used in the mirror equation to analyze image formation.

Convex Mirrors

Convex mirrors are curved outward and are known for their ability to form virtual, upright, and reduced images. These mirrors have a common property of reflecting a wider field of view, which is why they are often used in vehicles as side-view mirrors. Their image-producing characteristics are crucial in applications where a broad perspective is required, despite a reduction in image size.

Perception of Distance in Convex Mirrors

Because convex mirrors produce images that are smaller than the objects they represent, objects appear to be farther away than they really are. This optical effect can lead to a misjudgment of distances when relying solely on the mirror’s image, which is why warnings are often attached to convex mirrors. Understanding this concept is essential in applications where accurate perception of distance is critical, such as in vehicle safety.

Mirror Equation

The mirror equation, 1/f = 1/do + 1/di, relates the focal length (f) of a spherical mirror to the object distance (do) and the image distance (di). This formula is fundamental in geometrical optics and is used to determine the location and nature (real or virtual) of images formed by mirrors.

Magnification

Magnification in mirror systems is defined as m = -di/do, which also equals the ratio of the image height to the object height. It indicates how much larger or smaller the image is compared to the object, and the negative sign helps distinguish between an upright (positive) and an inverted (negative) image. This concept is essential when analyzing the properties of images formed by mirrors.

Transcript

00:02 In the given problem radius of curvature of the back view mirror in the car which is actually a convex spherical mirror its radius of curvature here is given as plus 18 .0 centimeter as in case of a convex mirror radius of curvature lies behind the mirror and here the sign conventions which we will be using all the distances in front of the mirror will be taken as negative and behind the mirror they will be taken as positive so its focal length also here f is equal to half of the radius of curvature means it will come out to be plus 9 .0 centimeter now in the first part of the problem there is another car behind this back view mirror which will serve as the object distance for this is spherical mirror.

01:17 So this object distance means s will come out to be minus 13 .0 centimeter.

01:31 This is meter which can also be written as minus 13 hundred centimeter.

01:40 Height of this object means height of the car that is given to us y is equal to 1 .5 meter or we can say this is 150 centimeter and that is also positive because the object is upright.

02:07 Now using the expression for lateral magnification which is actually the ratio of the height of the image to the height of the object and in terms of focal length this is given as f divided by f minus s.

02:28 So here is height of the image of the car will come out to be y into f divided by f minus s.

02:38 So plugging in all known values here, this y dash means height of the image of the car will be given by h will be given by y, which is 150 centimeter multiplied by focal length of the convex mirror, which is 9 centimeter divided by 9 centimeter again minus s, which is negative of 1300 centimeter...

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