Question

A convex mirror of radius of curvature 2 metres is attached to a motor bike to watch any other vehicle coming from behind. Calculate the location and position of a vehicle which is at 4 metres behind the bike as seen in the mirror. Also explain by the ray diagram.

Hint:

For convex mirrors, the image formed is always virtual, erect, and diminished. The image is formed behind the mirror at a distance smaller than the object distance.

Answer 1

We are given that the radius of curvature of the convex mirror is \( R = 2 \, \text{m} \). The focal length \( f \) of a mirror is related to the radius of curvature by the formula: \[ f = \frac{R}{2}. \] Substituting the given value of \( R \): \[ f = \frac{2}{2} = 1 \, \text{m}. \] The object distance \( u \) is given as 4 m (the distance of the vehicle behind the bike), so \( u = -4 \, \text{m} \) (we take it as negative for convex mirrors).
We use the mirror formula to calculate the image distance \( v \): \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u}. \] Substitute the known values of \( f \) and \( u \): \[ \frac{1}{1} = \frac{1}{v} + \frac{1}{-4}. \] Simplifying: \[ 1 = \frac{1}{v} - \frac{1}{4}. \] Solving for \( \frac{1}{v} \): \[ \frac{1}{v} = 1 + \frac{1}{4} = \frac{5}{4}. \] Thus, \[ v = \frac{4}{5} = 0.8 \, \text{m}. \] So, the image is formed at a distance of 0.8 m behind the mirror, meaning the vehicle appears to be 0.8 metres behind the mirror as seen in the mirror.
Conclusion:
The image of the vehicle is formed at a distance of 0.8 m behind the convex mirror, and the vehicle appears to be at a location of 0.8 m behind the mirror.