An object is placed against the center of a concave mirror and then moved along the central axis until it is 5.0 m from the mirror. During the motion, the distance $\left|i\right|$between the mirror and the image it produces is measured. The procedure is then repeated with a convex mirror and a plane mirror. Figure 34-28 gives the results versus object distance $\mathit{p}$. Which curve corresponds to which mirror? (Curve 1 has two segments.)

The expression for the mirror formula that is the relation of the image distance and object distance is as follows,

$\frac{\mathbf{1}}{\mathbf{f}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{v}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{u}}$ …(i)

Here,$f$ is the focal length,$v$ is the image distance, and$u$ is the object distance.

The mirror can be identified from the curves by using the properties of the mirrors. The straight line slope represents the graph of a plane mirror. While hyperbola graphs represent both the concave and convex mirrors being equally opposite graph behavior to each other.

According to the properties of mirrors, for flat mirror image distance is same as object distance. A concave mirror produces an image at infinity when the object is placed at the focal point.

Consider equation (i), and find the relation between the image and object distance.

$\left|i\right|=\frac{pf}{f-p}$

From the above properties, the following conclusions can be obtained.

Curve1corresponds to a concave mirror.

(a) It is known that the focal length of a plane mirror is infinity, thus using it in equation (i), a direct relationship between the object and image distance can be obtained, that is $v=u$.

(b) Curve3corresponds to a plane mirror because there is a direct proportion between the image distance and the object distance.

(c) Curve 2corresponds to a convex mirror.Consider equation (i), the relation between the image and object distance can be obtained as$\left|i\right|=\frac{pf}{p+f}$.The image is always virtual and exists. As the object distance p increases, the image distance$i$ also increase to approach a limiting value of $f$.

It is known that$p=\infty$, when the limit is obtained and it gives$i=f$

Thus, curve 1, 2, 3 correspond to concave, plane and convex mirrors respectively.