Problem Statement:
A particle is thrown upward from the bottom of an inclined plane. If it takes time t for the particle to move up and N×t to slide down to its initial position, we need to find the kinetic friction involved in this scenario.

Solution:
To solve this problem, we need to analyze the motion of the particle during its ascent and descent on the inclined plane.

1. Motion during ascent:
During the ascent, the particle moves against gravity and experiences a deceleration due to the gravitational force. The equation of motion during ascent is given by:
s = ut - 0.5gt^2

Where:
s = displacement
u = initial velocity (thrown upward)
g = acceleration due to gravity

Since the particle reaches its maximum height during ascent, its final velocity (v) is zero. Therefore, we can write:
v = u - gt = 0

Solving this equation, we get:
u = gt

2. Motion during descent:
During the descent, the particle moves with the force of gravity and experiences an acceleration due to gravity. The equation of motion during descent is given by:
s = ut + 0.5gt^2

Where:
s = displacement (initial position to final position)
u = initial velocity (at the topmost position)
g = acceleration due to gravity

Since the particle returns to its initial position, the displacement (s) is zero. Therefore, we can write:
0 = ut + 0.5gt^2

Solving this equation, we get:
u = -0.5gt

3. Finding the time taken for the descent:
From the given information, we know that the particle takes time t to move up and N times t to slide down to its initial position. Therefore, the time taken for descent is:
t_descent = N × t

4. Finding the initial velocity during descent:
Using the equation of motion during descent, we can substitute the value of t_descent and solve for u:
0 = u(N × t) + 0.5g(N × t)^2

Simplifying this equation, we get:
u = -0.5g(N × t)

5. Finding the kinetic friction:
The kinetic friction is the force acting against the motion of the particle on the inclined plane. It can be calculated using the equation:
Frictional force = mass × acceleration

Since the particle is moving down the inclined plane, the acceleration is due to gravity. Therefore, we can write:
Frictional force = mass × g

To find the mass of the particle, we can use the equation of motion during ascent:
s = ut - 0.5gt^2

Since the particle starts from the bottom of the inclined plane, the initial displacement (s) is zero. Therefore, we can write:
0 = ut - 0.5gt^2

Simplifying this equation, we get:
u = 0.5gt

Now, we can substitute the value of u in the equation for frictional force:
Frictional force = (0.5gt) × g = 0.5g^2t

Therefore, the kinetic friction involved in this scenario is 0.5g^2t.

Summary: