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A point object moves along the principal axis of a concave mirror of focal length 30 cm with speed 5 m/s toward the mirror. When the object is at a distance of 60 cm from the mirror, the speed and direction of the image are:

1. 5 m/s away from mirror
2. 5 m/s towards the mirror
3. 10 m/s away from mirror
4. 10 m/s towards the mirror

Correct answer: 5 m/s away from mirror

Solution

Concave mirror, f = -30 cm. Object at u = -60 cm. 1/v = 1/f - 1/u = 1/(-30) - 1/(-60) = -1/30 + 1/60 = -2/60 + 1/60 = -1/60. So v = -60 cm (real image, same side as object). Differentiating 1/v + 1/u = 1/f: -(1/v²)(dv/dt) - (1/u²)(du/dt) = 0 => dv/dt = -(v²/u²) * du/dt = -(v/u)² * du/dt. v/u = (-60)/(-60) = 1. dv/dt = -(1)² * (du/dt). Object moving toward mirror: du/dt = +5 m/s (u increasing from negative toward zero). Wait — u = -60 cm and object moves toward mirror means u increases (becomes less negative), so du/dt = +5 m/s. dv/dt = -(1)² * 5 = -5 m/s. v = -60 cm; dv/dt = -5 m/s means image moving toward mirror (v becoming more negative). But a negative v means image is in front of mirror. If dv/dt < 0, v is decreasing (becoming more negative), meaning image moves away from mirror... Sign convention: toward mirror is negative direction for u and v. Object velocity toward mirror = du/dt is positive (magnitude 5, and u becomes less negative). dv/dt = -5 m/s means image speed is 5 m/s away from mirror? Actually: dv/dt = -5 m/s means v is becoming more negative — since negative v means image is in front, more negative means farther from mirror. So image moves AWAY from mirror at 5 m/s. Answer: 5 m/s away from mirror.

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