Final answer:
The focal length of the lens can be determined using the lens formula and the given information about the object and image distances. By solving simultaneous equations and quadratic equations, we can find the values of v1 and v2 and calculate the focal length of the lens as 53.7 cm.
Step-by-step explanation:
To determine the focal length of the lens, we can use the lens formula:
1/f = 1/v - 1/u
Where f is the focal length, v is the image distance, and u is the object distance.
In this case, we are given that the lens forms two different images on the screen, separated by 20cm. Assuming the lens was not moved, we can consider that the object distance for the first image is 90cm and for the second image is 90 + 20 = 110cm (since the screen and object remained fixed).
Applying the lens formula for each image, we get:
1/f = 1/90 - 1/v1
1/f = 1/110 - 1/v2
We can solve these equations simultaneously to find the value of f. Subtracting the second equation from the first equation, we get:
1/90 - 1/110 = 1/v2 - 1/v1
Simplifying this, we get:
1/990 = (v1 - v2) / (v1 * v2)
Letting d = v1 - v2, this becomes:
1/990 = d / (v1 * v2)
Since d = 20cm, we can rewrite the equation as:
1/990 = 20 / (v1 * v2)
From this equation, we see that for a given product of v1 and v2, the value of f will be 990 times the product.
Therefore, to find the focal length, we need to determine the values of v1 and v2. We are given the distances of the images from the lens, but we need to convert them to v1 and v2. Let's say the distances of the first and second images from the lens are x and x + 20 respectively. Then, we have:
v1 = 90 - x and v2 = 110 - x
Substituting these values in the equation, we get:
1/990 = 20 / ((90 - x) * (110 - x))
Cross multiplying and simplifying, we get a quadratic equation:
x2 - 200x + 8910 = 0
Solving this equation, we find two possible values for x: x = 100 or x = 89.1
Since x cannot be negative and x + 20 must be less than 110 (to satisfy the given condition), we can conclude that the only valid solution is x = 89.1.
Substituting this value back into the equation for v1 and v2, we get:
v1 = 90 - 89.1 = 0.9 cm
v2 = 110 - 89.1 = 20.9 cm
Finally, substituting the values of v1 and v2 into the expression for the focal length, we find:
f = 990 / (0.9 * 20.9) = 53.7 cm
Therefore, the focal length of the lens is 53.7 cm.