Analysis of the Situation:
- Two blocks A and B are connected by a massless inextensible string and placed on a horizontal surface.
- The coefficient of friction between the block and the surface is given.
- A force F = kt, where k = 1 N/s and t is the time in seconds, is applied to block A.
- The acceleration of the system and the relative motion between the blocks will be determined.

Step 1: Free Body Diagrams
- Draw free body diagrams for both blocks A and B separately.
- Block A has force F applied in the right direction and friction force fA acting in the opposite direction.
- Block B experiences only the friction force fB acting in the opposite direction.

Step 2: Equations of Motion
- Apply Newton's second law of motion to both blocks individually.
- For block A, the net force is the difference between the applied force and the friction force: F - fA = mA * aA, where mA is the mass of block A and aA is its acceleration.
- For block B, the net force is the friction force: fB = mB * aB, where mB is the mass of block B and aB is its acceleration.

Step 3: Friction Force
- The friction force can be calculated using the equation: f = μ * N, where μ is the coefficient of friction and N is the normal force.
- The normal force is equal to the weight of the block: N = m * g, where m is the mass of the block and g is the acceleration due to gravity.
- Substituting the values, we get: fA = μ * mA * g and fB = μ * mB * g.

Step 4: Equations of Motion (Continued)
- Substituting the friction force expressions into the equations of motion, we have:
- F - μ * mA * g = mA * aA
- μ * mB * g = mB * aB
- Rearranging the equations, we get:
- aA = (F - μ * mA * g) / mA
- aB = μ * g

Step 5: Solution
- The acceleration of block A, aA, depends on the applied force F, the mass of block A, mA, and the coefficient of friction μ.
- The acceleration of block B, aB, depends only on the coefficient of friction μ and the acceleration due to gravity g.
- As time increases (t increases), the applied force F increases according to F = kt.
- Substituting the value of F into the equation for aA, we can calculate the acceleration of block A as a function of time.
- The relative motion between the blocks can be determined by comparing the accelerations of block A and block B.

Conclusion:
- By analyzing the given situation and applying the principles of Newton's laws of motion, we can determine the acceleration of each block and the relative motion between them.
- The acceleration of block A depends on the applied force, mass, and coefficient of friction, while the acceleration of block B depends only on the coefficient of friction and gravity.
- By calculating the values of aA and aB as functions of time, we can

Using force u can find acceleration as function of time , ..