Question

The image formed of an object at a distance of 25 cm from a convex mirror is half the length of the object. Determine (i) the distance of the image from the mirror, and (ii) the focal length of the mirror.

Hint:

The mirror formula \(\frac{1}{f} = \frac{1}{v} - \frac{1}{u} \) is used to determine focal length and image distance. For convex mirrors, both \( v \) and \( f \) are positive.

Answer 1

Using the mirror formula: \[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \] Given that the object distance \( u = -25 \) cm, and the magnification \( m = \frac{-v}{u} = \frac{1}{2} \), we can find \( v \) and then solve for \( f \). From magnification formula: \[ m = \frac{-v}{u} = \frac{1}{2} \Rightarrow v = \frac{-u}{2} = \frac{-(-25)}{2} = 12.5 \, \text{cm} \] Now, using the mirror formula with \( u = -25 \) cm and \( v = 12.5 \) cm: \[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} = \frac{1}{12.5} - \frac{1}{-25} \] Solving for \( f \), we get: \[ \frac{1}{f} = \frac{1}{12.5} + \frac{1}{25} = \frac{2}{25} \Rightarrow f = 25 \, \text{cm} \] Thus, the answers are: (i) \( v = 12.5 \, \text{cm} \) (ii) \( f = 25 \, \text{cm} \)