Introduction:
When a block slides down an inclined plane with a constant velocity and is then projected on the same plane with an initial velocity, it eventually comes to rest at a certain distance along the incline. In this explanation, we will determine how far the block will move before coming to rest.

Analysis:
To solve this problem, we need to consider the forces acting on the block and apply the principles of motion. Let's break down the analysis into the following key points:

1. Forces:
- Gravity force acts vertically downwards and can be resolved into two components: one parallel to the incline (mg sinθ) and the other perpendicular to the incline (mg cosθ).
- The normal force acts perpendicular to the incline and counterbalances the component of gravity perpendicular to the incline, i.e., mg cosθ.
- The frictional force acts parallel to the incline and opposes the motion of the block.

2. Sliding Down the Incline:
When the block slides down the inclined plane with a constant velocity, the frictional force exactly balances the component of gravity parallel to the incline. Therefore, the frictional force is given by:

Frictional force, f = mg sinθ

3. Projecting on the Incline:
When the block is projected on the incline with an initial velocity, it initially experiences an additional force due to the projection. This force is parallel to the incline and reduces with time until it becomes zero.

4. Coming to Rest:
To determine the distance the block will move before coming to rest, we need to find the point where the frictional force becomes zero. At this point, the net force acting on the block will be zero, causing it to come to rest.

5. Equilibrium Condition:
The net force on the block can be expressed as the difference between the component of the initial velocity parallel to the incline and the frictional force:

Net force = mg sinθ - μmg cosθ

where μ is the coefficient of friction.

6. Distance Calculation:
Equating the net force to zero, we can solve for the distance x:

mg sinθ - μmg cosθ = 0
μmg cosθ = mg sinθ
μ = tanθ

Therefore, the distance moved by the block before coming to rest is given by:

x = u² / (2g tanθ)

Conclusion:
In conclusion, the block will move a distance of u² / (2g tanθ) along the inclined plane before coming to rest. The analysis considered the forces acting on the block, the equilibrium condition, and the principles of motion to determine the distance.

V^2/2g