when a object is kept at a distance of 20cm from a cocave mirror,the image is formed at adistance of 10 cm .if the object is moved with a speedof 4m/s then the speed of image is?

To find the speed of the image formed by a concave mirror when the object is moved, we can use the mirror formula and the concept of magnification. Let's break this down step by step.

Understanding the Mirror Formula

The mirror formula for concave mirrors is given by:

1/f = 1/v + 1/u

Where:

• f = focal length of the mirror
• v = image distance (positive for real images)
• u = object distance (negative for real objects)

Given Values

In your scenario:

• Object distance, u = -20 cm (negative because it's in front of the mirror)
• Image distance, v = 10 cm (positive because it's a real image)

Finding the Focal Length

First, we can calculate the focal length using the mirror formula:

1/f = 1/v + 1/u

Substituting the values:

1/f = 1/10 + 1/(-20)

1/f = 0.1 - 0.05 = 0.05

Thus, f = 20 cm.

Calculating the Speed of the Image

Next, we need to determine how the image distance changes as the object moves. The speed of the image can be found using the relationship between the object distance and the image distance. The formula for the speed of the image (v_image) in relation to the speed of the object (v_object) is derived from the differentiation of the mirror formula:

v_image = - (v^2 / u^2) * v_object

Substituting the Values

We know:

• v = 10 cm
• u = -20 cm
• v_object = 4 m/s = 400 cm/s (converting to cm for consistency)

Now substituting these values into the formula:

v_image = - (10^2 / (-20)^2) * 400

v_image = - (100 / 400) * 400

v_image = - (1/4) * 400 = -100 cm/s

Interpreting the Result

The negative sign indicates that the image moves in the opposite direction to the object. Therefore, the speed of the image is 100 cm/s towards the mirror.

In summary, as the object moves towards the mirror at 4 m/s, the image moves at a speed of 100 cm/s in the opposite direction. This relationship highlights the dynamic nature of image formation in concave mirrors, where changes in object position directly affect image characteristics.