Karnataka Board PUCPUC Science 2nd PUC Class 12

A Small Block of Mass M and a Concave Mirror of Radius R Fitted with a Stand Lie on a Smooth Horizontal Table with a Separation D Between Them. the Mirror - Physics

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Sum

A small block of mass m and a concave mirror of radius R fitted with a stand lie on a smooth horizontal table with a separation d between them. The mirror together with its stand has a mass m. The block is pushed at t = 0 towards the mirror so that it starts moving towards the mirror at a constant speed V and collides with it. The collision is perfectly elastic. Find the velocity of the image (a) at a time t < d/V, (b) at a time t > d/V.

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Solution

Note :
(a) At time t = t,
   Object distance, u = −(d − Vt)
Here d > Vt , `f = -R/2` 

Mirror formula is given by:
`1/v = 1/f - 1/u = -2/R + 1/ (d-Vt)`

`=(-2 (d-Vt ) + R)/ (R ( d-Vt)`

`=(-R(d -V))/( R-2 ( d-Vt))`
Differentiating w.r.t. 't':

`dv/dt ={( -RV) [R-2( d-Vt)]-2V[R (d-Vt)] }/[R-2( d- Vt )]`

`=-R^2V/[2 (d-Vt)-R^2`
This is the required speed of mirror.
(b) When `t >d/2` the collision between the mirror and mass will take place. As the collision is elastic, the object will come to rest and the mirror will start to move with the velocity V.

`u =( d-Vt).`
At any time, `t >d/2 `
The distance of mirror from the mass will be:
`x = V ( t -d/v ) = Vt - d`
Here ,
Object distance, u `= -( Vt - d ) = ( d -Vt)`
Focal length, f `= -R/2`
now ' mirror formula is given by; 
`1/f = 1/v + 1/u`
`⇒ 1/v = 1/f - 1/u = 1/(R/-2) - 1/(d - Vt)`
` v ⇒ - [ (R(d-Vt) )/R - 2(d-Vt)]`

Velocity of image ( Vimage ) is given by,
`"Vimage" = (dv )/ (dt) = d/dt [ [R(d-Vt)} / (R+ 2 (d-Vt)]]`
if `y = d-Vt`
`dy/ dt =-V`

Velocity of image:
`'Vimage['=d/dt [ Ry/ R +2y ] = -Vr [[ R + 2y -2y ]/ ( R + 2y)^2] `

`=(-VR^2)/( R + 2y )^2`
As the mirror is moving with velocity V,
Therefore, 
V = Vimage + Vmirror
Absolute velocity of image `= V [ 1 - ( R^2) / ( R + 2y )^2]` 

`= V = [ 1 - ( R^2) / 2 ( Vt - d ) - R^2]`

Concept: Concave Mirror
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Chapter 18: Geometrical Optics - Exercise [Page 416]

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HC Verma Class 11, Class 12 Concepts of Physics Vol. 1
Chapter 18 Geometrical Optics
Exercise | Q 75 | Page 416

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