###### Advertisements

###### Advertisements

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*.

###### Advertisements

#### Solution

Note :

(a) At time *t* = *t*,

Object distance, *u* = −(*d* − V*t*)

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]`

#### APPEARS IN

#### RELATED QUESTIONS

An object is placed 15 cm in front of a convex lens of focal length 10 cm. Find the nature and position of the image formed. Where should a concave mirror of radius of curvature 20 cm be placed so that the final image is formed at the position of the object itself?

Suppose the lower half of the concave mirror's reflecting surface is covered with an opaque material. What effect this will have on the image of the object? Explain

A convex lens of focal length 20 cm is placed coaxially with a convex mirror of radius of curvature 20 cm. The two are kept 15 cm apart. A point object is placed 40 cm in front of the convex lens. Find the position of the image formed by this combination. Draw the ray diagram showing the image formation.

Obtain the mirror formula and write the expression for the linear magnification.

A convex lens of focal length 20 cm is placed coaxially in contact with a concave lens of focal length 25 cm. Determine the power of the combination. Will the system be converging or diverging in nature?

A convex lens of focal length 25 cm is placed coaxially in contact with a concave lens of focal length 20 cm. Determine the power of the combination. Will the system be converging or diverging in nature?

A convex lens of focal length f_{1} is kept in contact with a concave lens of focal length f_{2}. Find the focal length of the combination.

Find the diameter of the image of the moon formed by a spherical concave mirror of focal length 7.6 m. The diameter of the moon is 3450 km and the distance between the earth and the moon is 3.8 × 10^{5} km.

A particle goes in a circle of radius 2.0 cm. *A* concave mirror of focal length 20 cm is placed with its principal axis passing through the centre of the circle and perpendicular to its plane. The distance between the pole of the mirror and the centre of the circle is 30 cm. Calculate the radius of the circle formed by the image.

A concave mirror of radius R is kept on a horizontal table (See figure). Water (refractive index = μ) is poured into it up to a height h. Where should an object be placed so that its image is formed on itself?

A hemispherical portion of the surface of a solid glass sphere (μ = 1.5) of radius *r* is silvered to make the inner side reflecting. An object is placed on the axis of the hemisphere at a distance 3*r* from the centre of the sphere. The light from the object is refracted at the unsilvered part, then reflected from the silvered part and again refracted at the unsilvered part. Locate the final image formed.

The convex surface of a thin concavo-convex lens of glass of refractive index 1.5 has a radius of curvature 20 cm. The concave surface has a radius of curvature 60 cm. The convex side is silvered and placed on a horizontal surface as shown in figure. (a) Where should a pin be placed on the axis so that its image is formed at the same place? (b) If the concave part is filled with water (μ = 4/3), find the distance through which the pin should be moved so that the image of the pin again coincides with the pin.

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.

A gun of mass *M* fires a bullet of mass *m* with a horizontal speed *V*. The gun is fitted with a concave mirror of focal length *f* facing towards the receding bullet. Find the speed of separation of the bullet and the image just after the gun was fired.

A mass *m* = 50 g is dropped on a vertical spring of spring constant 500 N m^{−1} from a height *h* = 10 cm as shown in figure. The mass sticks to the spring and executes simple harmonic oscillations after that. A concave mirror of focal length 12 cm facing the mass is fixed with its principal axis coinciding with the line of motion of the mass, its pole being at a distance of 30 cm from the free end of the spring. Find the length in which the image of the mass oscillates.

Two concave mirrors of equal radii of curvature R are fixed on a stand facing opposite directions. The whole system has a mass m and is kept on a frictionless horizontal table following figure. Two blocks A and B, each of mass m, are placed on the two sides of the stand. At t = 0, the separation between A and the mirrors is 2 R and also the separation between B and the mirrors is 2 R. The block B moves towards the mirror at a speed v. All collisions which take place are elastic. Taking the original position of the mirrors-stand system to be x = 0 and X-axis along AB, find the position of the images of A and B at t = (a) `R/v` (b) `3R/v` (c) `5R/v`.