Halliday, Resnick & Walker — Images: Problem 69

69 (a) A luminous point is moving at speed \( v_O \) toward a spherical mirror with radius of curvature \( r \), along the central axis of the mirror. Show that the image of this point is moving at speed \[ v_I = -\left( \frac{r}{2p - r} \right)^2 v_O \] where \( p \) is the distance of the luminous point from the mirror at any given time. Now assume the mirror is concave, with \( r = 15 \text{ cm} \), and let \( v_O = 5.0 \text{ cm/s} \). Find \( v_I \) when (b) \( p = 40 \text{ cm} \) (far outside the focal point), (c) \( p = 7.8 \text{ cm} \) (just outside the focal point), and (d) \( p = 10 \text{ mm} \) (very near the mirror).