Here I challenge you with some fascinating math and physics problems that I have encountered, or thought of over the years. Note that I have selected the problems based on the enjoyment they offer in the solution process, rather than their difficulty. Enjoy them!

Problems

- Find the surface area and the volume of an \(n\)-dimensional sphere.
- Determine the number of degrees of freedom in an \(n\)-sided polygon.
- Imagine two identical objects with heat capacity \(C\) and temperatures \(T_1\) and \(T_2\). If we were to build an engine out of them, what is the maximum work that this engine can do? \(^1\)
- Give the shape of a fixed-volume planet (in mathematical form) that maximizes the gravitational force acting on a person standing on it.
- A ballistic missile is launched from the north pole, the target is at the latitude \(\phi\). To have the minimal launch velocity of the missile, at which angle (with respect to the horizon) do we need to launch it? \(^2\)
- Imagine a homogenous, liquid planet of average radius \(R \) and density \(\rho\), spinning around itself with an angular velocity of \(w.\) Find its eccentricity.
- A point particle of mass \(m\) and charge \(q\) is moving under a homogenous magnetic field of induction \(B\) and electrical field of intensity \(E\), directed perpendicular to each other. Ignoring gravity and relativity, find its trajectory.\(^1\)
- Imagine a water sprinkler in the garden, sprinkling water uniformly at each angle and in all directions at the same time. What radius from the origin would get watered the most?
- Consider a particle undergoing a one-dimensional random walk, taking steps of length \(a\) to the right or left with equal probability. What is the probability distribution of the particle's position after \(n\) steps?
- Consider a particle undergoing an \(n\)-dimensional random walk, taking steps in each direction with equal probability. What is the probability that the particle will come back to the origin?
- A person is collecting a set of distinct coupons, and each time they acquire a coupon, it is equally likely to be any one of the set. Determine the expected number of trials needed to complete the set.
- Calculate the repulsion force the light rays from the Sun apply onto the Earth.
- Describe the properties of a material that makes up a liquid planet, so that however much mass is removed, the radius of the planet stays the same.
- Find the oscillation period of a spherical ball oscillating in water.\(^1\)
- Two small particles with opposing charges \(q\) are separated by a distance \(L\) and left free. Find the collision time.\(^3\)
- A light source of intensity \(I\) and frequency \(f_0\) falls normally onto a mirror with mass \(m\) and initial velocity of zero. Find the frequency of the reflected beam as a function of time.
- A linearly polarised light beam transits from a medium with refraction index of \(n_1\) to another of \(n_2\), normal to the surface. Find the portion of the power that is reflected and refracted from the surface.\(^3\)
- Imagine that the clouds are made of tiny water droplets that hang onto the air and distributed homogenously in the space. Find the acceleration of a rain droplet that is falling down among them. The rain droplet, initially of negligible size, has a density of \(\rho\), and the water droplets in the cloud have a mean density of \(\lambda\). Assume the rain droplet absorbs all encountered water droplets while maintaining a spherical shape.
- Consider a rigid container from which all the air has been pumped out. Now, the valve of the container is slightly opened, and the air from outside is slowly filling in. The air flow stops when mechanical equilibrium is reached. Find the final temperature of the air inside. The room temperature is T, and the heat flux through the walls is ignored.\(^2\)
- How far away from the handle of a sword of length \(L\) should you hit an item, so that you feel the impact the least?
- Imagine (1) an object (2) light ray approaching a planet of mass \(M\) from infinity, with initial lateral distance to the center of the planet \(R\) and speed \(v\). Find the shift in the heading direction after it escapes the gravity of the planet.
- Prove the approximation \(n! \sim \sqrt{2\pi n} \left( \frac{n}{e} \right)^n\) (James Stirling, 1730)

References: \(^1\)Nahid Mammadov, \(^2\)Jaan Kalda, \(^3\)Rashadat
Gadmaliyev