Concave mirror forms a real image twice as large as the...

concave mirror forms a real image twice as large as the object. The object is then moved such that the new real image produced is three times the size of the object. If the image was moved $75 \mathrm{~cm}$ from its initial position, how far was the object moved and what is the focal length of the mirror?

Key Concepts

Mirror Equation

The mirror equation, given by (1/f) = (1/do) + (1/di), is a fundamental relation in geometrical optics that links a concave mirror's focal length (f) with the object distance (do) and the image distance (di). It allows us to determine one quantity when the other two are known and is essential for analyzing image formation in curved mirrors.

Magnification

Magnification is defined as the ratio of the image size to the object size, and in mirror problems it is also given by m = ?di/do. This concept not only indicates how large the image will appear compared to the object but also provides information about the orientation, with the negative sign indicating an inverted image in the case of real images from concave mirrors.

Real Image Formation

In concave mirrors, a real image is formed when the light rays physically converge after reflection. This occurs when the object is placed beyond the focal point of the mirror. Recognizing whether an image is real or virtual helps in applying the correct sign conventions and in understanding the geometric relationships involved in the mirror equation.

Object and Image Distance Relationships

The relationship between object distance and image distance is central to understanding image formation. Changes in the object position affect the image position, size, and nature of the image. In problems involving the movement of the object, an analysis of how these distances change helps to predict shifts in the image location, an essential consideration when using the mirror equation and magnification formulas together.

Transcript

00:02 Hello and welcome to this video solution of numerate.

00:05 Here we are given that a concave mirror forms a real image twice as large as the object right and the magnification m1 will be equal to 2.

00:14 Now it forms a real image that means the object and the image are in the same side right.

00:27 The expression of okay so you have got the mirror equation 1 by u plus 1 by v that is equal to 1 by f and the magnification is minus v by u right.

00:38 So when v and u are with same sign then this magnification will be negative.

00:42 So i take m1 to be negative and the object is then moved such that the new real image produces three times the size of the object.

00:51 So m2 will be equal to minus 3 now because it's again a real image.

00:58 Now it's given that the image was moved 75 centimeters from its initial position right.

01:04 So v2 will be equal to v1 plus 75.

01:07 Based on this you have to find out the focal length as well as the distance the object is moved.

01:14 So what you can do is you can start by let's say this equation from this mirror equation.

01:25 So you can plug in the value of u to be equal to minus v by m or here 1 by f will be equal to minus m by v plus 1 by v.

01:44 I take v1 m1 that will be equal to minus m2 by v2 plus 1 by v2.

01:51 Now using this equation you can calculate v1...

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