(a) A luminous point is moving at speed vs toward a spherical mirror with radius of curvature r,...

Question

(a) A luminous point is moving at speed vs 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

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 cm, and let v_o = 5.0 cm/s. Find v₁ when (b) p = 30 cm (far outside the focal point), (c) p = 8.0 cm (just outside the focal point), and (d) p = 10 mm (very near the mirror).

Robert · Accepted Answer

(a)We have 1/f=1/p+1/i. This is the Gaussian form of the thin lens formula. 
1/i=1/f-1/p 
1/i=(p-f)/pf 
Substituting f=r/2 we get, 
i=(pf)/(p-f)=[p(r/2)]/(p-(r/2)) 
i=pr/(2p-r) 
Differentiating both sides w.r.to time we get, 
v1=di/dt=d/dt(pr/(2p-r)) 
Substituting v0=-dp/dt we get,

(a) A luminous point is moving at speed vs 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 where p is...

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