A convex mirror with a radius of curvature of 3 m is used as a rearview mirror for a vehicle. If a bus is located at 5 m from this mirror, find the position, nature, and size of the image.
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We have a convex mirror with radius of curvature 3 meters used as a rearview mirror. A bus is located 5 meters from the mirror. We need to find the position, nature, and size of the image formed by this convex mirror.
First, we identify the given values. The radius of curvature is 3 meters, and the object distance is negative 5 meters since the object is in front of the mirror. For a convex mirror, the focal length equals radius divided by 2, which gives us f equals 1.5 meters. The focal length is positive for convex mirrors.
Now we apply the mirror formula: one over f equals one over u plus one over v. Substituting our values: one over 1.5 equals one over negative 5 plus one over v. Solving for v: one over v equals two thirds plus one fifth, which equals thirteen fifteenths. Therefore, v equals fifteen thirteenths, or approximately 1.15 meters.
Now we calculate the magnification using the formula m equals negative v over u. Substituting our values: m equals negative fifteen thirteenths divided by negative 5, which simplifies to three thirteenths, or approximately 0.23. Since m is positive, the image is erect. Since the absolute value of m is less than 1, the image is diminished. Since v is positive, the image is virtual and located behind the mirror.
To summarize our solution: The image is located 1.15 meters behind the convex mirror. It is virtual and erect in nature. The image is diminished with a magnification of 0.23. Convex mirrors always produce virtual, erect, and diminished images, making them perfect for rearview mirrors as they provide a wider field of view.