Simulate planetary orbits using Newton's law of gravitation $F = \frac{Gm_1m_2}{r^2}$ and explore Kepler's laws of planetary motion. Visualize elliptical trajectories, orbital velocity changes, and the relationship between orbital period and semi-major axis.
Apply calculus to motion analysis using $v = \frac{dx}{dt}$ and $a = \frac{dv}{dt}$, and integrate acceleration to find velocity and position functions. Visualize how derivatives and integrals connect position, velocity, and acceleration graphs in real-time.
Calculate moment of inertia $I = \int r^2 dm$ for various geometric shapes and apply the parallel axis theorem $I = I_{cm} + Md^2$. Visualize how mass distribution affects rotational inertia and compare common shapes like disks, spheres, and rods.
Analyze rolling motion with the no-slip condition $v_{cm} = R\omega$ and explore energy partitioning between translational and rotational kinetic energy. Visualize objects rolling down inclines and see how moment of inertia affects acceleration.
Calculate the center of mass for multi-particle systems using $\vec{r}_{cm} = \frac{\sum m_i\vec{r}_i}{\sum m_i}$ and analyze 2D collision dynamics. Visualize how momentum conservation applies independently to x and y components in elastic and inelastic collisions.