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.