Velocity of image of moving object in spherical mirror

(a) Velocity components along the axis:
From mirror formula,
(1/u) + (1/v) = 1/f
Differentiating the above formula on both sides with time, we get

(vim)∥ = Velocity of image w.r.to mirror along the principle axis
(vom)∥ = Velocity of object w.r.to mirror along the principle axis
Velocity component perpendicular to principle axis:
m = hi/ho ⇒ hi = mho
Differentiating the above equation on both sides w.r.to time, we get

(vim)⊥ = Velocity of image w.r.to mirror perpendicular to the principle axis
(vom)⊥ = Velocity of object w.r.to mirror perpendicular to the principle axis
= rate of change of magnification
m = – V/U
By differentiating this equation on both sides w.r.to time, we get

POSITION AND NATURE OF IMAGE FORMATION IN SPHERICAL MIRROR
Image formation in concave mirror:

S.No Position of object Position of image Nature of image
1 At ∞ At F Real and inverted image,| m | << 1.
2 Between C and ∞ Between F and C Real and inverted image,| m | < 1.
3 At C At C Real and inverted, | m | = 1.
4 Between F and C Between C and ∞ Real and inverted, | m | > 1.
5 At F At infinity Real and inverted, | m | >> 1.
6 Between F and P Behind the mirror Virtual and erect, | m | > 1.

Image formation in convex mirror:

S.no Position of object Position of image Nature of image
1 At ∞ At F Virtual and erect, | m | << 1.
2 Anywhere between ∞ and P Between P and F Virtual and erect, | m | < 1.

Position, size and nature of image formed by the spherical mirror

Mirror Location of the object Location of the image Magnification, size of the image Nature
Real Virtual Erect inverted
(a) Concave
At infinity, i.e., u = ∞ At focus, i.e., v = f m << 1, diminished Real inverted
Away from center of curvature (u > 2f) Between f and 2f, i.e., f < v < 2f m < 1, diminished Real inverted
At center of curvature u = 2f At center of curvature, i.e., v = 2f m = 1, same size as that of the object Real inverted
Between center of curvature and focus: F > u > 2f Away from the center of curvature v > 2f m > 1,magnified Real inverted
At focus, i.e,. u = f At infinity, i.e., v = ∞ m = ∞, magnified Real inverted
Between pole and focus u < f v > u m > 1,magnified virtual erect
(b) Convex
At infinity, i.e., u = ∞ At focus, i.e., v = f m < 1, diminished virtual erect
Anywhere between infinity and pole Between pole and focus m < 1, diminished virtual erect
Linear magnification Areal magnification
Transverse Longitudinal
When a object is placed perpendicular to the principle axis, then linear magnification is called lateral or transverse magnification.
It is given by

(*Always use sign convention while solving the problems) 		When object lies along the principle axis then its longitudinal magnification

If object is small;
Also length of image
= (v/u)2 × Length of object (L0)

If a 2D – object is placed with its plane perpendicular to principle axis
Its areal magnification

⇒ ms = m2 = Ai/A0