Experienced Math tutor | Researcher in theoretical physics | Pure and Applied mathematics| Elementary to advanced level |

The Student and Tutor practiced solving differential equations using Euler's numerical method. They implemented the method in Python, comparing the approximate solution to an exact one and observing the impact of step size on accuracy. The Student was assigned a homework problem to apply the method to a different differential equation.

The session began with the Student asking for help with a Python programming problem involving iterating through a data frame to find specific value sequences. The main part of the class focused on advanced physics concepts related to rocket science and fluid dynamics, covering topics such as multi-stage rockets, orbit trajectory, nozzle design principles (subsonic vs. supersonic flow), laminar and turbulent flow, and the various forces affecting a rocket's flight (thrust, gravity, drag, Magnus force). The Tutor also introduced the Navier-Stokes equation and discussed that numerical methods like Euler and Runge-Kutta will be explored in the next class for solving these complex equations.

The Tutor and Student discussed core principles of rocket propulsion, including the impact of nozzle design on thrust, fluid dynamics (laminar vs. turbulent flow, Mach paradox, Navier-Stokes equation), and the derivation of the rocket equation. They explored the practical necessity of multi-stage rockets for achieving Earth's escape velocity and briefly reviewed historical and current space exploration missions.

The class focused on modeling energy dissipation in particle systems, explaining how to introduce friction through resistive backgrounds and inelastic collisions. The tutor then derived equations for particle collisions using conservation of energy and momentum, providing formulas for calculating final velocities and discussing their application in simulations and rocket propulsion.

The tutor and student reviewed concepts of kinematics, including calculating velocity and acceleration from positional data over discrete time intervals. They then explored applying kinematic equations to simulate motion, particularly in the context of particle collisions and the effects of gravity, with a plan to introduce friction in future sessions.

The Tutor and Student delved into the principles of projectile motion and the derivation of kinematic equations using calculus and vector decomposition. They explored simulating trajectories and discussed methods for handling collisions in computational physics, with plans to focus on 2D collision scenarios in the next session.