Question

A concave mirror ha a magnification of -2.5, if its focal length is 20cm, find its image and object position​

Answer 1

Given :

• magnification of concave mirror, m = - 2.5
• focal length of concave mirror, f = -20 cm

[ focal length of concave mirror is always negative ]

To find :

• position of object, u = ?
• position of object, v = ?

Knowledge required :

• Formula for magnification of mirror

$\red{\bigstar}\boxed{\sf{m=\dfrac{-v}{u}}}$

• Mirror formula  

$\red{\bigstar}\boxed{\sf{\dfrac{1}{f}=\dfrac{1}{v}+\dfrac{1}{u}}}$

[ where m is magnification , v is position of image, us is position of object , f is focal length of mirror ]

Solution :

Finding relation between position of image and position of object

Using formula for magnification

$\implies\sf{m=\dfrac{-v}{u}}$

$\implies\sf{-2.5=\dfrac{-v}{u}}$

$\implies\sf{2.5=\dfrac{v}{u}}$

$\implies\red{\sf{v=2.5\;u\;\;\;\;\;\;eqn(1)}}$

Calculating position of image and position of object

Using mirror formula

$\implies\sf{\dfrac{1}{f}=\dfrac{1}{v}+\dfrac{1}{u}}$

$\implies\sf{\dfrac{1}{-20}=\dfrac{1}{2.5\;u}+\dfrac{1}{u}}$

$\implies\sf{\dfrac{1}{-20}=\dfrac{1+2.5}{2.5\;u}}$

$\implies\sf{\dfrac{1}{-20}=\dfrac{3.5}{2.5\;u}}$

$\implies\sf{u=\dfrac{3.5\times -20}{2.5}}$

$\implies\boxed{\red{\sf{u=-28\;cm}}}$

using equation (1)

$\implies\sf{v=2.5\;u}$

$\implies\sf{v=2.5\times-28}$

$\implies\boxed{\red{\sf{v=-70\;cm}}}$

Hence,

• position of object is 28 cm infront of mirror.
• position of image is 70 cm infront of mirror.

Answer 2

◘ Given ◘

$\bigstar$ Magnification of concave mirror, M = -2.5

$\bigstar$ Focal length of concave mirror, f = -20 cm

◘ To Find ◘

The image position(v) and object position​(u).

◘ Solution ◘

We know, for a concave mirror,

► Magnification, M = -v/u

Substituting the values :-

→ -2.5 = -v/u

→ 2.5 = v/u

→ v = 2.5u                 . . . (i)

____________________

Applying mirror formula :-

1/v + 1/u = 1/f

→ 1/2.5u + 1/u = 1/(-20)

→ (1 + 2.5)/2.5u = -1/20     [from (i)]

→ 3.5/2.5u = -1/20

→ 2.5u/3.5 = -20

→ 2.5u = -20 × 3.5

→ 2.5u = -70

→ v = -70 cm                [from (i)]

____________________

Putting the value of 'v' in equation (i) :-

-70 = 2.5u

→ -70 = 2.5 × u

→ u = -70/2.5

→ u = -28 cm

Therefore, the position of object is 28 cm in front of the mirror. and the  position of image is 70 cm in front of mirror.