Preamble The classical mathematician practically by instinct views the continuous process as the “real” process, and the discrete process as an approximation to it. The mathematics of finance and certain topics in the modern theory of stochastic processes suggest that, in some cases at least, the opposite is true. Continuous processes are, generally speaking, the…
Introduction In a previous article on measuring battery internal resistance, a simple technique for low-resistance measurement was outlined. In this article, the technique has been modified and expanded somewhat although the basic method remains the same. It is about as simple as it gets – one passes current through a reference resistor in series with…
Introduction A commonly encountered school-level Physics practical is the determination of the internal resistance of a battery – typically an AA or D cell. Typically this is based around a simple model of such a cell as a source emf in series with a small resistor. The cell is connected to a resistive load and…
Introduction Whilst no doubt most frequenters of “Physics Forums” will be familiar with Nicolaus Copernicus as the scientist who advanced the (at the time) radical thesis that the earth revolved around the sun rather than vice versa, a perhaps less well-known aspect of his work is the “nuts and bolts” geometry underlying his ground-breaking treatise:…
Introduction Every secondary school student who has encountered trigonometry in his/her Math syllabus will most likely have come across the sine, cosine, and area rules which are typically used to solve triangles in which certain information is supplied and the remainder are to be calculated. Somewhat surprisingly (because it is relatively simple to derive), the…
Introduction In a previous Physics Forums article entitled “How to Master Projectile Motion Without Quadratics”, PF user @kuruman brought to our attention the vector equation ##\frac{|V_0 \times V_f|}{g} = R## and lamented the fact that: “Equally unused, untaught and apparently not even assigned as a “show that” exercise is Equation (4) that identifies the range as the…
Introduction A recent homework problem that appeared in the forums was concerned with maximizing the horizontal range of a projectile subject to the launch site being a fixed height above the ground upon which the projectile was eventually impacted. Several interesting methods of solution arose so the idea of this article is to present all…
Introduction Being a somewhat geeky Maths ‘nerd’, I spent days leading up to Valentine’s Day trying to find a Maths function appropriate to the day. In the end, I did not find an appropriate single function but (perhaps aptly enough) a pair: function and inverse function. It occurred to me that these provide a good…
Introduction In any school Physics course, the Newtonian equations of motion are very much a ‘stock’ item. Students learn the equations and are given a variety of problems that provide practice in determining which equation(s) to use to solve any particular problem. What is perhaps a little surprising is that in general, no one applies…
Introduction In this article, we take as our starting point the original equations which Compton drew up and solved in his ground-breaking 1925 article: From the above equations, Compton solved for two variables namely ##\beta## the ratio of electron speed to the velocity of light, and for ##\nu_{\theta}##, the frequency of the scattered…
Introduction In a previous article entitled “Alternate Approach to 2D Collisions” we analyzed collisions between a moving and stationary object by defining the coordinate axes as being respectively parallel and perpendicular to the post-collision direction of motion of the stationary object. In this article, we will be adopting the same approach to analyze the well-known…
Introduction This article follows on from the previous on an alternate approach to solving collision problems. In that article, we determined the equal and opposite collision impulse to have magnitude ##\mu \Delta v## for perfectly inelastic collisions, ##\mu(1+e) \Delta v## for semi-elastic collisions, and ##2\mu \Delta v## for elastic collisions which will be the focus…
Introduction This article is essentially an addition to the previous one on (mainly) inelastic collisions to include the particular case of inelastic relativistic collisions. Reasons for writing a separate article are first that this author is not particularly well qualified to write on the topic and so may well need to request the scrutiny of…
Introduction Collisions are very much a stock item in any school physics curriculum and students are generally taught about the use of the principles of conservation of momentum and energy for solving simple collision problems in one dimension. In this article, we will be examining a very common type of collision problem: the inelastic collision….
Introduction In previous articles relating to various transition energies in Hydrogen, Helium, and Deuterium we have employed the following formula for electron energy given a particular primary quantum number n: $$ E_{n}=\mu c^2\sqrt{1-\frac{Z^2\alpha^2}{n^2}} $$ where ## \alpha ## is the fine structure constant and ## \mu ## the reduced electron mass for a single electron…
Introduction In this article, we will be revisiting a somewhat understudied (and seemingly unrepeated) experiment to measure the Deuterium Lyman Alpha line at approximately 121.5 nm. The experiment was carried out in the 1950s in the wake of the Lamb-Retherford experiment (1947) which established the tiny energy shift (Lamb shift) between the Hydrogen (and Deuterium)…
Introduction We are in many ways a fortunate generation having so many wonderful tools at our disposal on account of the “silicon revolution”. The advent of powerful personal computers has radically changed just about every form of human activity in a few short decades (this author – for example – is of a generation that…
Introduction In a previous article “Calculating the Balmer Alpha Line” we mentioned how accurate predictions of the spectral lines of singly ionized Helium were of considerable importance in persuading the scientific community that Danish physicist Niels Bohr was on the right track in respect of his groundbreaking atomic model first published in 1913. In this…
Introduction Most readers acquainted with the hydrogen spectrum will be familiar with the set of lines in the visible spectrum representing transitions of electrons from energy levels 3,4,5 and 6 (H alpha, beta, gamma, and delta respectively) of atomic hydrogen to energy level 2 – the Balmer series lines. The picture below shows 3 of…