An object is kept at rest on the principal axis of a lens. Initially the object is at a distance three times the focal length `'f'` of the lens. The lens runs towards the object at a constant speed `u,` until the distance between the object and its real image becomes `4 f`. If the iamge always forms on a moving screen then express the velocity of the screen as a function of time.

Explanation

To solve the problem step by step, let's break it down:

### Step 1: Understand the Initial Setup
The object is placed at a distance of \(3f\) from the lens, where \(f\) is the focal length of the lens. The lens moves towards the object at a constant speed \(u\).

### Step 2: Determine the Distance Between the Object and the Lens
As the lens moves towards the object, the distance between the lens and the object at time \(t\) can be expressed as:
\[ d = 3f - ut \]
Here, \(d\) is the distance from the lens to the object at time \(t\).

### Step 3: Use the Lens Formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
Where:
- \(u\) is the object distance (which we have as \(d\)),
- \(v\) is the image distance.

Substituting \(d\) into the lens formula gives:
\[ \frac{1}{f} = \frac{1}{3f - ut} + \frac{1}{v} \]

### Step 4: Rearranging the Lens Formula
Rearranging the lens formula to find \(v\):
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{3f - ut} \]
Combining the fractions:
\[ \frac{1}{v} = \frac{(3f - ut) - f}{f(3f - ut)} = \frac{2f - ut}{f(3f - ut)} \]
Thus, we can express \(v\) as:
\[ v = \frac{f(3f - ut)}{2f - ut} \]

### Step 5: Differentiate to Find Velocity of the Image
To find the velocity of the image \(v_i\), we differentiate \(v\) with respect to time \(t\):
\[ \frac{dv}{dt} = \frac{d}{dt}\left(\frac{f(3f - ut)}{2f - ut}\right) \]
Using the quotient rule:
\[ \frac{dv}{dt} = \frac{(2f - ut)(-fu) - f(3f - ut)(-u)}{(2f - ut)^2} \]
Simplifying this expression gives:
\[ \frac{dv}{dt} = \frac{-fu(2f - ut) + fu(3f - ut)}{(2f - ut)^2} \]
This simplifies to:
\[ \frac{dv}{dt} = \frac{fu(f)}{(2f - ut)^2} \]

### Step 6: Express the Velocity of the Screen
Since the image is always formed on a moving screen, the velocity of the screen \(v_s\) is equal to the velocity of the image:
\[ v_s = \frac{fu^2}{(2f - ut)^2} \]

### Final Result
Thus, the velocity of the screen as a function of time is:
\[ v_s = \frac{fu^2}{(2f - ut)^2} \]