A particle travels at a constant speed V from a large distance towards a concave mirror of radius R , moving along the mirror's principal axis. Determine the speed of the image formed by the mirror as a function of the particle's distance x from the mirror.

Step-by-step solution

The mirror formula is given by: 1 v + 1 u = 1 f Differentiating with respect to time t : - 1 v^2 dv dt - 1 u^2 du dt = 0 dv dt = - ( v u )^2 du dt The speed of the object is V , so | du dt | = V . The speed of the image is given by v_i = | dv dt | = m^2 V , where m = - v u is the transverse magnification. From the mirror formula, the magnification can be expressed as: m = f f - u For a concave mirror, the focal length is f = - R 2 and the object distance is u = -x . Substituting these values: m = - R 2 - R 2 - (-x)