@?school

Flashcards 2022-01-18 What are the four characteristics of a planet (ORFC)?? It is in __O__rbit around a star It has a mass large enough that gravity makes it __R__ound It has no __F__usion reactions It has __C__leared its orbit of most other objects What is the difference between a planet and a dwarf planet?? A dwarf planet has not cleared its neighbourhood. What is an asteroid?? An small, uneven and un-icy object that orbits the sun.

Flashcards 2022-01-18 What is the first stage in the evolution of a star?? A nebula. What is the main element in a nebula?? Hydrogen. What happens to a nebula after a long period of time?? It collapses under its own gravity. As a nebula is collapsing under its own gravity, what happens to the stores of energy?? The gravitational potential potential is transfered to kinetic energy. What happens as the gravitational energy of the dust particles in a nebula is transferred to kinetic energy?

See Also [[Maths - Numerical Methods]]S Try out an interactive visualisation of Euler’s method here: Euler’s method. Flashcards 2021-12-08 How could you summarise Euler’s method for solving first-order differential equations?? Start with some point on the curve and then follow the direction of the curve. If a gradient is given by $\frac{\text{d}y}{\text{d}x}$, how much would you increase the $y$-coordinate for a step size of $h$?? $$ y_1 = y_0 + \frac{\text{d}y}{\text{d}x} h $$

Flashcards 2021-12-08 What is gravitational potential?? The energy transferred per unit mass to move an object from infinity to a point. What is the gravitational potential at infinity?? $$ 0\text{J}\text{kg}^{-1} $$ Why is gravitational potential always negative?? Because you’re having to do the opposite of what gravity wants you to do. What is gravitational potential energy?? The energy transferred to move an object from infinity to a point. What does the graph of $V$ against $r$ look like for gravitational potential?

Flashcards 2021-12-06 What speed must a satellite be going at to orbit a planet at radius $r$?? $$ \sqrt{\frac{GM}{r}} $$ What three things must be true for a geostationary satellite?? It must be in an orbit above the Earth’s equator. It must rotate in the same direction as the Earth’s rotation. It must have an orbital period of 24 hours. What is true about a polar orbit?? It passes over a planet’s poles.

See Also [[Further Maths - L'Hôpital's Rule]]S [[Further Maths - Taylor Series]]S Flashcards 2021-12-05 How would you rewrite $$\lim_{x \to \infty} \frac{2-3x}{1+x}$$ in order to evaluate it without L’Hôpital’s rule?? $$ \frac{\lim_{x \to \infty} 2 - 3x}{\lim_{x \to \infty} 1 + x} $$ How could you evaluate $$\lim_{x \to \pi/2} (x-\frac{\pi}{2})\tan x$$ using a Taylor series?? Approximate $\cot x$ around $\pi/2$ What must you make sure to do when evaluating a limit with a Taylor series?

Flashcards 2021-12-05 In what form can you express $a\sin x \pm b\cos x$?? $$ R\sin(x \pm \alpha) $$ In what form can you express $a\cos x \pm b\sin x$?? $$ R\cos(x \mp \alpha) $$ You want to express $$a\sin x \pm b\cos x$$ as $$R\sin(x \pm \alpha)$$ How can you calculate $R$?? $$ R = \sqrt{a^2 + b^2} $$ You want to express $$a\sin x \pm b\cos x$$ as $$R\sin(x \pm \alpha)$$ How can you calculate $\alpha$?

See Also [[Further Maths - Maclaurin Series]]S [[Further Maths - Limits]]S Flashcards 2021-12-01 What is the Maclaurin series a special case of?? The Taylor series. What is the formula for the Taylor series about $x = a$?? $$ f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + … $$ When is the Taylor series valid for $x = a$?? When $f^{(n)}(a)$ exists and is finite for all natural numbers and for values of $x$ for which the infinite series converges.

Flashcards 2021-12-01 What is the Weierstrass substitution?? The substitution $t = \tan\frac{x}{2}$ used to evaluate integrals. What do you substitute for $\text{d}x$ in the Weierstrass substitution?? $$ \text{d}x = \frac{2}{1 + t^2} \text{d}t $$ Why is the Weierstrass substitution useful?? Because it turns complicated integration of trig functions into a rational function. What technique would you use for evaluating $$\int^{\pi/2}_{pi/3} \frac{1}{1 + \sin x - \cos x} \text{d}x$$?? The Weierstrass substitution. $$\int \csc (x) \text{d}x$$ Can you make the Weierstrass substitution?

Flashcards 2021-12-01 What do Kepler’s laws describe?? The motion of orbits. What is Kepler’s First Law?? All of the planets move in elliptical orbits with the sun at one focus. What is Kepler’s Second Law?? The radius vector sweeps out equal areas in equal times. What is the “radius vector” for Kepler’s Second Law?? The imaginary line between the sun and the planet. Which of Kepler’s laws does this show?? Kepler’s second law. What is Kepler’s Third Law?

These are my notes for studying Maths, Further Maths, Computing and Physics at A-level. These notes weren’t written with anyone else in mind, so if they do not make a whole lot of sense or seem strange you might be lacking the context that I had at the time of writing them. There’s also a lot of spelling mistakes (see the Further Maths ‘matricies’ topic for example). For a list of all the notes for each subject, see:

Flashcards 2021-11-26 How could you formulate the travelling salesperson problem in terms of Hamiltonian cycles?? Finding the shortest Hamiltonian cycle on a graph. For small graphs, how can you solve the travelling salesperson problem?? Try every Hamiltonian cycle. What is the nearest neighbour algorithm used for?? Finding a Hamiltonian cycle that can be used as an upper bound for the TSP. What are the 4 steps of the nearest neighbour algorithm?? Choose any start vertex Go to the nearest vertex that hasn’t been included Repeat 2 until all vertices have been included Return directly to the start vertex What would be the first step in using the nearest neighbour algorithm on this distance matrix?