Question

A particle is moving at a constant speed V from a large distance towards a concave mirror of radius R along its principal axis. Find the speed of the image formed by the mirror as a function of the distance x of the particle from the mirror.

Answer 1

Given,:

Radius of the concave mirror = R

 f=R/2

Velocity of the particle,

V=dx/dt

Object distance, u = −x

Using mirror equation,

1/v+1/u=1/f

On substituting the  values we get,

1/v+1/-x= -2/R

1/v= -2/R+1/x =R-2x/Rx

V=Rx/(R-2x)

Velocity of the image is given by V1

V1=dv/dt=d/dt [Rx/(R-2x)

=d/ dt [ Rx/( R-2x) ] - d/dt [(R-2x)Rx] /( R-2x)²

=R [ (dx/ dt) (R-2x) ] - 2(dx/dt)x / (  R-2x)²

=R[V (R-2x) -[2V×0] / (R-2x)²

=VR²/(2x-R)²

Answer 2

ʀᴀᴅɪᴜs ᴏғ ᴛʜᴇ ᴄᴏɴᴄᴀᴠᴇ ᴍɪʀʀᴏʀ = ʀ

 ғ=ʀ/2

ᴠᴇʟᴏᴄɪᴛʏ ᴏғ ᴛʜᴇ ᴘᴀʀᴛɪᴄʟᴇ,


ᴠ=ᴅx/ᴅᴛ

ᴏʙᴊᴇᴄᴛ ᴅɪsᴛᴀɴᴄᴇ, ᴜ = −x

ᴜsɪɴɢ ᴍɪʀʀᴏʀ ᴇǫᴜᴀᴛɪᴏɴ,


1/ᴠ+1/ᴜ=1/ғ

ᴏɴ sᴜʙsᴛɪᴛᴜᴛɪɴɢ ᴛʜᴇ  ᴠᴀʟᴜᴇs ᴡᴇ ɢᴇᴛ,


1/ᴠ+1/-x= -2/ʀ

1/ᴠ= -2/ʀ+1/x =ʀ-2x/ʀx

ᴠ=ʀx/(ʀ-2x)

ᴠᴇʟᴏᴄɪᴛʏ ᴏғ ᴛʜᴇ ɪᴍᴀɢᴇ ɪs ɢɪᴠᴇɴ ʙʏ ᴠ1


ᴠ1=ᴅᴠ/ᴅᴛ=ᴅ/ᴅᴛ [ʀx/(ʀ-2x)

=ᴅ/ ᴅᴛ [ ʀx/( ʀ-2x) ] - ᴅ/ᴅᴛ [(ʀ-2x)ʀx] /( ʀ-2x)²

=ʀ [ (ᴅx/ ᴅᴛ) (ʀ-2x) ] - 2(ᴅx/ᴅᴛ)x / (  ʀ-2x)²

=ʀ[ᴠ (ʀ-2x) -[2ᴠ×0] / (ʀ-2x)²

=ᴠʀ²/(2x-ʀ)²