A small object is approaching a convex mirror of
focal length 15 cm along its principal axis with a
speed 16.2 cm/s. The speed of the image when
the object is 12 cm away from the mirror is
Answers
focal length of convex mirror, f = -15 cm
speed of object with respect to mirror, du/dt = 16.2 cm/s
object distance, u = -12 cm
using formula,
1/v + 1/u = 1/f
⇒1/v + 1/-12 = 1/-15
⇒1/v = 1/-15 + 1/12
⇒1/v = (-12 + 15)/180 = 3/180
⇒1/v = 1/60
⇒v = 60cm
differentiating formula both sides with respect to time,
-1/v² × (dv/dt) - 1/u² × (du/dt) = 0
⇒(dv/dt) = -(v/u)² × (du/dt)
now putting values of , v , u and du/dt
⇒dv/dt = -(60cm/12cm)² × (16.2 cm/s)
= -25 × (16.2 cm/s)
= -402 cm/s
hence, speed of image with respect to mirror is 402 cm/s and it is directed in just opposite direction of speed of object with respect to mirror.
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focal length of convex mirror, f = -15 cm
speed of object with respect to mirror, du/dt = 16.2 cm/s
object distance, u = -12 cm
using formula,
1/v + 1/u = 1/f
⇒1/v + 1/-12 = 1/-15
⇒1/v = 1/-15 + 1/12
⇒1/v = (-12 + 15)/180 = 3/180
⇒1/v = 1/60
⇒v = 60cm
differentiating formula both sides with respect to time,
-1/v² × (dv/dt) - 1/u² × (du/dt) = 0
⇒(dv/dt) = -(v/u)² × (du/dt)
now putting values of , v , u and du/dt
⇒dv/dt = -(60cm/12cm)² × (16.2 cm/s)
= -25 × (16.2 cm/s)
= -402 cm/s