The problem involves a block moving along a smooth curved path, then transitioning to a rough inclined plane where kinetic friction, denoted by $\mu_k = 0.2$, slows it down until it stops at a height $y_{\text{maks}}$, as shown in the image.

The question is asking for the equation that represents the final position $y_{\text{maks}}$ of the object after stopping on the inclined plane.

Key Points:

• Kinetic Friction: $\mu_k = 0.2$
• Inclined plane angle: $\theta$
• Initial height: $h$
• Energy considerations: The initial energy is purely potential, and the energy loss is due to friction as the object moves up the inclined plane.

Solving the Problem:

1. Initial Energy:

   • The object starts from rest at height $h$, so its potential energy is $PE = mgh$.

2. Energy Loss due to Friction:

   • As the object moves up the incline, friction does work against the motion.
   • The work done by friction is $W_f = \mu_k mg \cos \theta \cdot d$, where $d$ is the distance traveled up the incline.

3. Final Position:

   • At the maximum height $y_{\text{maks}}$, the object stops, meaning all of the remaining energy is potential energy at $y_{\text{maks}}$, which is $PE = mg y_{\text{maks}}$.
   • Using the principle of conservation of energy: Initial potential energy = Final potential energy + Work done by friction.

   The equation becomes: $mgh = mg y_{\text{maks}} + \mu_k mg \cos \theta \cdot d$ where $d$ can be related to $y_{\text{maks}}$ through the incline's geometry: $d = \frac{y_{\text{maks}}}{\sin \theta}$.

4. Solving for $y_{\text{maks}}$: After substituting $d$ and simplifying, we obtain the expression for $y_{\text{maks}}$: $y_{\text{maks}} = \frac{h}{1 + \mu_k \cot \theta}$

Conclusion:

Based on the above analysis, the correct option is:

• Answer: D.

Would you like further clarification or a deeper breakdown of any part of the solution?

Here are 5 related questions to explore the topic:

1. How does the angle of incline affect the final height $y_{\text{maks}}$?
2. What role does the coefficient of friction $\mu_k$ play in determining the stopping point of the object?
3. How would the result change if there were no friction on the inclined plane?
4. What if the object were to start with some initial kinetic energy? How would the equation change?
5. How do we derive the relationship between $d$ and $y_{\text{maks}}$ using trigonometry?

Tip: When dealing with energy problems, always account for both potential and kinetic energy, and consider the work done by external forces like friction.