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Perform the maths showing proper steps with accurate numerical calculations

### Results and Discussion

The plots show the velocity of the stone as a function of time for both friction models, using Euler and Runge-Kutta methods:

#### Case i: \( p(v) = -0.1v \) (Linear Friction)
- The velocity steadily decreases due to the constant gravitational pull and linear drag force. Both methods closely match, with only slight differences between Euler and Runge-Kutta solutions.

#### Case ii: \( p(v) = 0.1v^2 \) (Quadratic Friction)
- In this case, the quadratic friction slows down the object faster, causing the velocity to reach negative values more quickly. The quadratic term amplifies the deceleration as velocity increases. Again, Runge-Kutta is slightly more accurate than Euler for larger step sizes.

Both methods provide consistent results with the analytical understanding of friction's impact on the motion. The Euler method shows a slightly more significant numerical error due to its simpler approximation, while the Runge-Kutta method, being higher-order, is more precise.

Would you like more detailed step-by-step numerical values for these methods or a deeper analysis of one of the cases?

### 5 Related Questions:
1. How would changing the mass of the stone affect the velocity results in both cases?
2. What impact does decreasing the step size \( h \) have on the accuracy of both numerical methods?
3. How would the results differ if the frictional force depended on \( v^3 \) instead of \( v \) or \( v^2 \)?
4. Can you derive the analytical solution for the differential equation in either case and compare it with the numerical results?
5. What real-world systems could be modeled with similar equations but more complex forces?

**Tip**: For more accurate results with numerical methods, decreasing the step size \( h \) can significantly reduce errors, but it increases computational cost.

Solve two problems related to numerical methods for a falling stone and a boundary value problem using finite difference and Euler/Runge-Kutta methods.

Learn how to solve a falling stone problem using Euler's and Runge-Kutta methods for different frictional forces, and tackle a boundary value problem using the finite difference method. This detailed solution provides step-by-step numerical methods for solving differential equations.

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