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.
Download our app for free and get started
Given that, u = distance of the object = -x
f = focal length $=-\frac{\text{R}}{2}$
and, V = velocity of object $=\frac{\text{dx}}{\text{dt}}$
From mirror equation, $\frac{1}{-\text{x}}+\frac{1}{\text{v}}=-\frac{2}{\text{R}}$
$\frac{1}{\text{v}}=-\frac{2}{\text{R}}+\frac{1}{\text{x}}=\frac{\text{R}-2\text{x}}{\text{R}\text{x}}\Rightarrow\text{v}=\frac{\text{Rx}}{\text{R}-2\text{x}}=$ Image distance
So, velocity of the image is given by,
$\text{V}_1=\frac{\text{dv}}{\text{dt}}=\frac{\Big[\frac{\text{d}}{\text{dt}}(\text{xR})(\text{R}-2\text{x})\Big]-\Big[\frac{\text{d}}{\text{dt}}(\text{R}-2\text{x})(\text{xR})\Big]}{(\text{R}-2\text{x})^2}$
$=\frac{\text{R}\Big[\frac{\text{dx}}{\text{dt}}(\text{R}-2\text{x})\Big]-\Big[-2\frac{\text{d}}{\text{dt}}\text{x}\Big]}{(\text{R}-2\text{x})^2}=\frac{\text{R}[\text{v}(\text{R}-2\text{x})+2\text{vx}0}{(\text{R}-2\text{x})^2}$
$=\frac{\text{VR}^2}{(2\text{x}-\text{R})^2}=\frac{\text{R}\big[\text{VR}-2\text{xV}\big)+2\text{xV}}{(\text{R}-2\text{x})^2}$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1The refractive index of a material $M_1$ changes by 0.014 and that of another material $M_2$ changes by 0.024 as the colour of the light is changed from red to violet. Two thin prisms one made of $M_1(A = 5.3^\circ)$ and other made of $M_2(A = 3.7^\circ)$ are combined with their refracting angles oppositely directed.View Solution
- Find the angular dispersion produced by the combination.
- The prisms are now combined with their refracting angles similarly directed. Find the angular dispersion produced by the combination.
- 2Consider the situation shown in figure. The elevator is going up with an acceleration of $2.00m/s^2$ and the focal length of the mirror is 12.0cm. All the surfaces are smooth and the pulley is light. The mass-pulley system is released from rest (with respect to the elevator) at t = 0 when the distance of B from the mirror is 42.0cm. Find the distance between the image of the block B and the mirror at $t = 0.200s$. Take $g = 10m/s^2$.View Solution
- 3View SolutionA lady uses +1.5D glasses to have normal vision from 25cm onwards. She uses a 20D lens as a simple microscope to see an object. Find the maximum magnifying power if she uses then microscope
- Together with her glass
- Without the glass. Do the answers suggest that an object can be more clearly seen through a microscope without using the correcting glasses?
- 4View SolutionAn object 1.5 cm in size is placed on the side of the convex lens in the arrangement (a) above. The distance between the object and the convex lens is 40 cm. Determine the magnification produced by the two-lens system, and the size of the image.
- 5View SolutionA point object is placed at a distance of 15cm from a convex lens. The image is formed on the other side at a distance of 30cm from the lens. When a concave lens is placed in contact with the convex lens, the image shifts away further by 30cm. Calculate the focal lengths of the two lenses.
- 6View SolutionOne day Chetan’s mother developed a severe stomach ache all of a sudden. She was rushed to the doctor who suggested for an immediate endoscopy test and gave an estimate of expenditure for the same. Chetan immediately contacted his class teacher and shared the information with her. The class teacher arranged for the money and rushed to the hospital. On realising that Chetan belonged to a below average income group family, even the doctor offered concession for the test fee. The test was conducted successfully.
Answer the following questions based on the above information:- Which principle in optics is made use of in endoscopy?
- Briefly explain the values reflected in the action taken by the teacher.
- In what way do you appreciate the response of the doctor on the given situation?
- 7Three thin prisms are combined as shown in figure. The refractive indices of the crown glass for red, yellow and violet rays are $\mu_\text{r},\mu_\text{y}$ and $\mu_\text{v}$ respectively and those for the flint glass are $\mu'_\text{r},\mu'_\text{y}$ and $\mu'_\text{v}$ respectively. Find the ratio $\frac{\text{A}'}{\text{A}}$ for which.View Solution
- There is no net angular dispersion.
- There is no net deviation in the yellow ray.
- 8View SolutionIn many experimental set-ups the source and screen are fixed at a distance say D and the lens is movable. Show that there are two positions for the lens for which an image is formed on the screen. Find the distance between these points and the ratio of the image sizes for these two points.
- 9View Solution
- Draw a ray diagram to show refraction of a ray of monochromatic light passing through a glass prism.
- Explain briefly how the phenomenon of total internal reflection is used in fibre optics.
- 10View Solution
- Draw the ray diagram showing refraction of light through a glass prism and hence obtain the relation between the refractive index $\mu$ of the prism, angle of prism and angle of minimum deviation.
- Determine the value of the angle of incidence for a ray of light travelling from a medium of refractive index $\mu_1=\sqrt{2}$ into the medium of refractive index $\mu_2=1$, so that it just grazes along the surface of separation.