Reflection Class - XII Part - 4

Relation between object and image velocity:

1. When object is moving along the principal axis:

Let us consider, a point object is moving along the principal axis of a concave mirror of focal length f. At any instant the object distance is u and image distance is v. Using mirror formula we get, ——- (1).

Differentiating equation (1) with respect to time we get, + =

Or, – – = 0 [f is constant so, = 0]

Or, = – —— (2)

Here (= v_o) is the rate at which u is changing. This represents the object velocity with respect to the mirror. Similarly, (= v_i) represents the image velocity with respect to the mirror. If m is the magnification, then m = – .

Therefore, = – m²

So, v_i = – m²v_o

The negative sign indicates that when u decreases v increase simultaneously.

2. When object is moving normal to the principal axis:

Let us considered, a point object O is at a distance u on the principle axis of a concave mirror. The corresponding image is formed at point I at a distance v from the pole of the mirror.

As the object is moving along a straight line perpendicular to the principal axis, then there is no change in the object and image distance.

At any instant of time, the position of object is at point O^/ at a distance x from the principal axis. At that time the corresponding image I^/ is at a distance y from the principle axis of mirror.

Magnification m

Or, m =

Therefore, y = mx

Differentiating the equation with respect to time we get, = m .

Here (= v_o) represents the object velocity with respect to the mirror. Similarly, (= v_i) represents the image velocity with respect to the mirror.

Hence, v_i = mv_o.

Example: A point object is moving with velocity u at an angle with principle axis towards a stationary concave mirror of focal length f. Find the velocity of image when the object is at a distance of 1.5f from the mirror.