Question #dafd5

Let at an instant the image distance be `v` for the real object distance `u` during movement of a point object towards a convex mirror of focal length `F` along its axis.

We know that the cojugate foci relation of spherical mirror is as follows.

`color(blue)(1/v+1/u=1/F......[1])`

Imposing sign convention for convex mirror
`F->+ve and u=-u" for real object"` we get equation [1] as

`color(green)(1/v-1/u=1/F.....[2])`

`color(green)(=>1/v=1/u+1/F)`

`color(green)(=>1/v=(u+F)/(uF))`

`color(green)(=>v/u=(F)/(u+F))`

`color(green)(=>v/u=(F)/(u+F)<1.....[3])`

Now differentiating equation [2] w.r to `t` we get

`color(violet)(-1/v^2(dv)/(dt)+1/u^2(du)/(dt)=0.....[4])`

`color(violet)(=>1/v^2(dv)/(dt)=1/u^2(du)/(dt))`

Now `(dv)/(dt)=v_i="speed of image"`

And `(du)/(dt)=v_o="speed of object"`

So equation [4] becomes

`color(red)(1/v^2xxv_i=1/u^2xxv_o)`

`color(red)(=>v_i/v_o=v^2/u^2.....[5])`

Utilising equation [3] and [5] we can say

`color(red)(v_i/v_o=F^2/(u+F)^2)`

So `color(red)(v_i < v_o)" when "abs(u)< abs(F) or abs(u)>abs(F)`

And `color(red)(v_i =v_o)" when "abs(u)< abs(F) and abs(u)->0`

Since `lim_(u->0)(F^2/(u+F)^2)=1`

So we can support option (1) and (3) to be correct