Problem: A concave mirror of radius of curvature two meters is placed at the bottom of a tank of water. The mirror forms an image of the sun when it is directly overhead. Calculate the distance of the images from the mirror for 1. 160 cm and 2. 80 cm of water in the tank.

Solution:
To solve this problem, we can use the mirror formula which relates the object distance (u), image distance (v), and focal length (f) of a mirror. The formula is given by:

1/f = 1/v - 1/u

In this case, the mirror is concave, so the focal length is negative. The radius of curvature (R) is given as 2 meters, which means the focal length (f) is half of the radius of curvature.

Step 1: Calculate the focal length (f)
Given the radius of curvature (R) is 2 meters, we can calculate the focal length (f) using the formula:
f = R/2 = 2/2 = 1 meter

Step 2: Calculate the object distance (u)
The object distance (u) is the distance between the mirror and the object, which is the height of the water in the tank.

For Case 1: 160 cm of water
u = 160 cm = 1.6 meters

For Case 2: 80 cm of water
u = 80 cm = 0.8 meters

Step 3: Calculate the image distance (v)
Using the mirror formula:
1/f = 1/v - 1/u

For Case 1: 160 cm of water
1/1 = 1/v - 1/1.6

Simplifying the equation, we get:
1/v = 1 - 1/1.6
1/v = 0.625

Taking the reciprocal of both sides, we find:
v = 1/0.625 = 1.6 meters

For Case 2: 80 cm of water
1/1 = 1/v - 1/0.8

Simplifying the equation, we get:
1/v = 1 - 1/0.8
1/v = 0.25

Taking the reciprocal of both sides, we find:
v = 1/0.25 = 4 meters

Step 4: Conclusion
The distance of the image from the mirror for Case 1 (160 cm of water) is 1.6 meters, and for Case 2 (80 cm of water) is 4 meters.

Summary:
- Focal length (f) of the concave mirror is calculated as half of the radius of curvature (R).
- The object distance (u) is the height of the water in the tank.
- The image distance (v) is calculated using the mirror formula 1/f = 1/v - 1/u.
- For Case 1 (160 cm of water), the image distance is 1.6 meters.
- For Case 2 (80 cm of water), the image distance is 4 meters.