For anyone interested in the physics: Qualitatively, the result is similar.
Qualitatively, the best angle decreases rapidly the higher the friction is:
Thank you to everyone who replied here!
I found the error. It's not a mathematical or numerical curiosity, but a simple coding error. Namely,
np.absolute(vel)
does not give \vert \dot{\mathbf x} = \sqrt{\dot{x}^2+\dot{y}^2}, as I thought it would, but (\vert \dot x \vert, \vert \dot y \vert...
The standard definition used in mathematics (in fields such as calculus of variations, 'convex analysis' or 'convex optimization') is that a function is convex if its epigraph \operatorname{epi}f := \lbrace (x, y) : y \geq f(x) \rbrace is a convex set. For a twice differentiable function, this...
Hey sorry, I've made a sign error. The two friction terms compensate each other, not add up. Actually not surprising: Without gravity, motion is along a straight line.
I will correct it. Correct is
d^2 y / dx^2 = - \frac{1}{2 \dot{x}^2}
Still, it's concave. In fact, it is *always* concave...
1/\dot{x}^2 \geq 0 manifestly, so it remains to check that -2 c \dot{y} \sqrt{\dot{x}^2+\dot{y}^2}-\frac 1 2 \leq 0 in order for d^2 y/ d x^2 \leq 0 (i.e. concavity). But -2 c \sqrt{\dot{x}^2 +\dot{y}^2} is always negative, so if \dot{y} \geq 0, everything is negative.
Hey!
This started very harmless... A friend and I were throwing stones in a lake. Mine didn't get very far, he was teasing me "What was the ideal angle again?". Of course, I know it should be 45°. I replied in jest: "That's because I'm considering air resistance!" Then we had a discussion what...
When I was in your age, I liked Jay Orear's Physics and the Feynman lectures a lot. In general: You cannot learn physics without math. So if you want to learn physics, you must learn math first, namely vectors+matrices, analysis (differentiation and integration) and later vector calculus (the...
Some information about Germany: (I'm German and I have studied at the universities of Heidelberg and Munich)
It is not true that German universities have no tuition. It is true that the tuition is very very low, so low that it will never be your primary concern, but the costs of living will be...
Hi everyone!
I've got a question about recognition chances of my German degrees at Canadian grad schools. Ideally there would be people out there who did their Bachelor+Master in Germany or in a country with a similar academic system (3+2 years, Master as a precondition to Ph.D. rather than an...
Sorry, I don't know how to make a proper quote.
As to your first question: I wrote proton, not photon. :) I was wondering whether I can just draw a Feynman diagram leaving the proton's quarks as they are. For me, it made no sense to write down +p on both the left and the right side of the...
Aaah! That's why the diagrams are always drawn with curved lines! I was wondering about that the whole semester.
Thank you so much! That was a very valuable insight for me!
I believe I have something to add. Do I simply have uud \to uud and the \gamma \to u \overline{u} individually? Or should I take one up quark (or down, respectively) from the proton? Because if I would just leave the proton as it is then it would not at all make sense to me to mention it in...
Hi everyone!
In an exam on particle physics there has been a problem where for a number of decays we were asked to either reason their non-existence (i.e. name a conservation law that it contradicts to) or draw a Feynman diagram. However, with one of those decays I have a problem:
\gamma + p...