Problem 3 - Master Resource Book in JEE Main Physics

Question

When an object is kept at a distance of \(30 \mathrm{~cm}\) from a concave mirror, the image is formed at a distance of \(10 \mathrm{~cm}\). If the object is moved with a speed of \(9 \mathrm{~ms}^{-1}\), the speed with which images moves, is (a) \(0.1 \mathrm{~ms}^{-1}\) (b) \(1 \mathrm{~ms}^{-1}\) (c) \(3 \mathrm{~ms}^{-1}\) (d) \(9 \mathrm{~ms}^{-1}\)

Step-by-Step Solution

Answer

The image moves with a speed of \(1 \text{ ms}^{-1}\), option (b).

Step 1: Understand the mirror formula

The mirror formula relates the object distance \( u \), the image distance \( v \), and the focal length \( f \) of a mirror: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]. In this problem, \( u = -30 \) cm (object distance is negative for mirrors), and \( v = -10 \) cm (since the image formed by a concave mirror for a real object is real and inverted, hence \( v \) is also negative).

Step 2: Calculate the focal length

Using the mirror formula: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} = \frac{1}{-10} + \frac{1}{-30} \]. Simplifying, we get \( \frac{1}{f} = \frac{-3}{30} + \frac{-1}{30} = \frac{-4}{30} = \frac{-2}{15} \). Therefore, \( f = -\frac{15}{2} = -7.5 \) cm.

Step 3: Use the magnification formula to relate speeds

The magnification \( m \) is given by \( m = \frac{-v}{u} \) and in terms of velocities \( \frac{v_{\text{image}}}{v_{\text{object}}} = \frac{v^2}{u^2} \). Substituting the initial given condition values, \( \frac{(-10)^2}{(-30)^2} = \frac{100}{900} = \frac{1}{9} \).

Step 4: Calculate the speed of the image

Given the object speed \( v_{\text{object}} = 9 \text{ ms}^{-1} \), use the magnification relation, \( v_{\text{image}} = \frac{1}{9} \times 9 = 1 \text{ ms}^{-1} \).

Key Concepts

Mirror Formula

The mirror formula is a fundamental equation in optics used for calculating image distance, object distance, and focal length in curved mirrors, such as concave mirrors. It is expressed as:

• \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \)

where:

• \( f \) is the focal length of the mirror
• \( v \) is the image distance
• \( u \) is the object distance

In this equation, all distances are measured from the mirror's surface. For concave mirrors, real object distances \( u \) and real image distances \( v \) are negative according to sign conventions. This formula helps predict where an image will form for given object and focal lengths. Understanding this relationship is crucial for working with optics in practical scenarios.

Image Distance

The image distance \( v \) is the distance between the mirror and the image formed by it. In this exercise, the object is at 30 cm away from the mirror, and the image distance given is 10 cm. Since we are dealing with a concave mirror, and the formed image is real and inverted, both \( u \) and \( v \) are negative in the mirror formula. Here, image distance \( v \) is -10 cm.
This negative sign indicates that the image appears on the same side as the object, a characteristic of real images produced by concave mirrors. Understanding how the image distance behaves helps in determining the size, nature, and position of the image.

Object Speed

The object speed refers to how fast the object is moving towards or away from the mirror. In this scenario, the object moves with a speed of 9 m/s. When dealing with changes in speeds between objects and their images in mirrors, it is necessary to use the concept of magnification.
Magnification, in terms of velocity, helps in calculating the image speed based on the object's speed. Using the relation \( \frac{v_{\text{image}}}{v_{\text{object}}} = \frac{v^2}{u^2} \), we can figure out how quickly the image moves in relation to the object's movement. This transformation of speeds is practical in real-world applications like designing optical instruments.

Focal Length

Focal length \( f \) is an essential characteristic of the concave mirror that describes how strongly the mirror converges or diverges light. It is the distance from the mirror's reflective surface to its focal point, where parallel light rays converge after reflection. In our problem scenario, the focal length is calculated using the mirror formula:

• Plugging \( v = -10 \) cm and \( u = -30 \) cm into \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \)
• We find \( \frac{1}{f} = \frac{-4}{30} \), leading to \( f = -7.5 \) cm

The negative value reflects that the mirror is concave, which focuses light to a point at the focal length distance. The focal length offers insights into image characteristics and plays a role in calculating image positions and sizes.