Ulysses’ trip

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  • When a supersonic airplane flies over your head

    Suppose a supersonic airplane has Mach number $M$. It flies horizontally. At some time, it flies past over your head at height $h$. Then, the distance between you and it when you have just heard it is $Mh$.

  • Finding curvature radius physically

    Sometimes the curvature radius of a curve can be found by using physical methods although it seems that you must use calculus to find it. In this article, the curvature radius of the curve $x^2=2py$ at the point where the curvature radius is smallest is found by using physical methods without using calculus (with only high school knowledge). The answer is that the smallest curvature is exactly $p$, and the point with smallest curvature is the vertex.

  • A whirling point charge

    In the vacuum, inside a fixed ring of radius $R$ with fixed charge $Q$ uniformly distributed, there is a point charge with charge $q$ and mass $m$ moving in the plane of the ring due tue the electrostatic force. It moves in the small region around the center of the ring, and the motion is periodic along a closed curve. The area of the region enclosed by the curve is $S$. Denote the distance from the center to the point charge as $r$, and $r\ll R$. Find the magnetic induction $B$ at the center of the ring.

  • Solving ODE by recursive integration

    By recursively integrating according to $x_{n+1}\!\left(t\right):=\int_{t_0}^tf\!\left(x_n\!\left(s\right),s\right)\,\mathrm ds+C$ from $x_0\!\left(t_0\right):=C$, we can get the solution of the ODE $x’\!\left(t\right)=f\!\left(x\!\left(t\right),t\right)$ with initial conditions $x\!\left(t_0\right)=C$ as the limit of the sequence of functions.

  • An example of non-uniform elements: heavy elastic rope

    To illustrate the concept about non-uniform elements, we study a simple problem: suppose a uniform heavy elastic rope has mass $m$, original length $L_0$, and stiffness $k$, and find the mass distribution and length of it when hung vertically. We can use the element method to solve this problem, but the elements are non-uniform in terms of length. The elements add up to get the total length $L=\frac{mg}{2k}+L_0$.

  • Kinetic energy, momentum, and angular momentum of rigid bodies

    In this article, we will find that the inertia matrix naturally appears when we calculate the kinetic energy $T$ or the angular momentum $\mathbf M$ of a rigid body. Then, we introduce the concept of principal inertia $\mathbf J_{\mathrm{pri}}$. We also study how the inertia matrix changes under translations and rotations and how those transformations may lead to conclusions that can help us simplify the calculation of inertia matrices.

  • Hölder means inequality

    The Hölder mean of $\vec x$ with weights $\vec w$ and a parameter $p$ is defined as $M_{p,\vec w}\!\left(\vec x\right):=\left(\vec w\cdot\vec x^p\right)^{\frac 1p}$, and the value at $p=-\infty,0,+\infty$ are defined by the limits. We can prove using Jensen’s inquality that the Hölder mean increases as $p$ increases. This property can be used to prove HM-GM-AM-QM inequalities.

  • A polynomial whose sum of coefficients is a factorial

    The function $\left(1-z\right)^{n+1}\sum_{k=1}^\infty k^nz^k$ is a polynomial of degree $n$ w.r.t. $z$, and the sum of its coefficients is $n!$. This turns out to be properties of Eulerian numbers.

  • The image of a circular object through a thin lens

    The image of a circle with radius $r$ and centered at $C\left(-2f,0\right)$ through a thin lens at $x=0$ with focal length $f$ and centered at $O\left(0,0\right)$ is a conic section with the focus being $\left(2f,0\right)$, the directrix being line $x=f$, and the eccentricity being $\frac rf$.

  • The point on the circle farthest to two lines

    Suppose $P$ is a point on the circle $\odot C$. When is the sum of distances from $P$ to two edges of $\angle O$ extremal? It turns out to be related to angle bisectors (the intersections of $\odot C$ and the bisector of $\angle O$ or its adjacent supplementary angle are extremals), while the edge cases (at the intersections of $\odot C$ and edges of $\angle O$) are a little tricky: we need to use the bisectors to divide the plane into four quadrants, pick the two quadrants where the line intersecting $\odot C$ at $P$ lies, translate the region to make it center at $C$, and see whether $P$ is inside the translated region.

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