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607.


Problem 44.4 (RHK)


A luminous point is moving at speed toward a spherical mirror, along its axis. (a) We have to show that the speed at which the image of this point object is moving is given by

Assuming that the mirror is concave, with and that , we have to find the speed of the image (b) if the object is far outside the focal point ; (c) if it is close to the focal point ; and (d) if it is very close to the mirror .


Solution:           Click For PDF Version

Relation between the object distance, o, image distance, , and the radius of curvature, r, for a spherical mirror is given by the equation

,

where we use the sign convention of Halliday Resnick and Krane.

We note from the spherical mirror equation that

Therefore, the speed of the image, , is the time derivative of the image distance .

We thus find

(a)

We are given that , and .

If the object is far outside the focal point ,

(b)

If the object is close to the focal point

(c)

If the object is very close to the mirror ,