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The mirror formula is a key relationship in optics that helps us find unknown distances related to mirrors. Specifically, it connects the focal length ( f ), the image distance ( v ), and the object distance ( u ). It is given by the equation: \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \). This equation applies to both concave and convex mirrors and is essential for determining one of these variables if the other two are known.
For a convex mirror, the image distance is considered negative because the image is formed behind the mirror. The same goes for the focal length, which also takes a negative value due to the mirror's shape. Understanding the mirror formula allows us to solve many optical problems, such as determining where an object must be placed to form an image at a certain distance.

Image distance ( v ) in mirror optics refers to the distance between the mirror and the position where the image appears. For convex mirrors, the image is always virtual, smaller, and upright—it forms behind the mirror, resulting in a negative image distance in calculations.
In our exercise, an object is given to have a v of -7 cm. This negative sign indicates the image is virtual and forms behind the convex mirror. In optic problems, understanding the sign convention is crucial as it directly affects the application of the mirror formula and results interpretation.

• Image distance is always negative for convex mirrors.
• A virtual image forms where light rays appear to converge.
• Use the mirror formula for solving for image distance.

The focal length ( f ) of a mirror is the distance from the mirror to its focal point, where light rays parallel to the principal axis converge (or appear to diverge in the case of convex mirrors). It is a crucial parameter for all mirrors in optics and directly influences image formation.
For convex mirrors, the focal length is negative. This is due to the focal point being behind the mirror in these types of mirrors. In our example, the focal length is calculated as \(-\frac{14}{3}\) cm. Knowing the focal length allows us to use the mirror formula effectively as it links to both the image and object distances.

• Focal length is negative in convex mirrors.
• Determines curvature and power of mirrors.
• Vital in calculating image and object distances.

Magnification ( m ) in optics describes how much larger or smaller an image is compared to the object itself. It is defined as the ratio of the height of the image ( h' ) to the height of the object ( h ), and also as the negative ratio of image distance to object distance: \( m = \frac{-v}{u} \).
In convex mirrors, the magnification is positive, indicating that the image is upright and virtual. In our scenario, the magnification from the first object was calculated using \( \frac{-7}{-14} \), giving \( \frac{1}{2} \). When magnification equals 1, the image and object are the same height—here, it is less than 1, meaning the image is smaller.

• Convex mirrors have positive magnification for upright images.
• Useful for comparing sizes between image and object.
• Makes calculating changes in image size simple.

Object distance ( u ) is the distance from the mirror to the point where the object is placed. In convex mirrors, this is typically positive, as real objects are in front of the mirror. It's an important value for determining image characteristics using the mirror formula.
In our case, the initial object distance was 14 cm. The challenge was to find a new object distance for an object with a different height leading to the same image height. By understanding the relationship between object and image distances through the mirror formula, the focus is on finding how changing u affects v .

• Essential for using the mirror formula effectively.
• Positively impacts real-world applications and solutions.
• Focuses on how distance and image characteristics correlate.