A short object of length L is placed along the principal axis of a concave mirror away from focus. The object distance is u. If the mirror has a focal length f, what will be the length of the image? You may take L<<|v-f|.

Hint: Use the mirror formula.

Step 1: Find the image of the ends of the object.

Since the object distance is u. Let us consider the two ends of the object to be at distance u1=u-L/2 and u2=u+L/2, respectively so that u1-u2=L. Let the image of the two ends be formed at v1 and v2, respectively so that the image length would be
                        L'=v1-v2                  ...(i)
Applying mirror formula, we have,
                                                   1u+1v=1f or v=fuu-f
On solving, the positions of two images are given by,
                                                          v1=f (u-L/2)u-f-L/2, v2=f(u+L/2)u-f+L/2

Step 2: Find the length of the image.
For length, substituting the value in (i), we have,
                                                           L'=v1-v2=f2L(u-f)2-L2/4
Since, the object is short and kept away from the focus, we have.
                                                             L2/4<<(u-f)2
Hence, finally,                                 L'=f2(u-f)2L
This is the required expression of the length of the image.