Recent content by tomwilliam

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    Combinatorics question -- Probability of being dealt a certain hand from a deck of cards

    Thank you! I considered this possibility, but discarded it on the following logic: If the question means that the sequence has to be all of the same suit, it should still be a factor of 4 in answer (C), as there are four possible suits for it to work with. If the question wants a specific...
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    Combinatorics question -- Probability of being dealt a certain hand from a deck of cards

    A quick translation of the question: We have a deck of 40 cards, containing four suits (hearts, diamonds, spades, clubs), in which each suit has an ace, a queen, a jack, a king and six number cards (2 to 7). From the deck, six cards are distributed randomly and successively to a player who picks...
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    I Derivative using the chain rule

    Thanks to both of you...I understand it now.
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    I Derivative using the chain rule

    Ok, thanks, I see where my mistake came in. So now I have ## \frac{d\sqrt{u}}{du}=(1/2)u^{-1/2} ## and ## \frac{du}{d\epsilon}= 2(y' + \epsilon g') ## so ## \frac {df}{d\epsilon} = (1/2)u^{-1/2}\times 2(y' + \epsilon g') ## but I seem to be a factor of ##g'## out.
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    I Derivative using the chain rule

    I'm coming back to maths (calculus of variations) after a long hiatus, and am a little rusty. I can't remember how to do the following derivative: ## \frac{d}{d\epsilon}\left(\sqrt{1 + (y' + \epsilon g')^2}\right) ## where ##y, g## are functions of ##x## I know I should substitute say ##u = 1...
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    A How do I define Open cluster membership?

    Sorry - I think I wasn't clear in my original post. I'm not looking for the definition of an open cluster, which you've summarised well, but I'm trying to define* membership of that cluster. I am studying some of the properties of a few open clusters, as identified by other authors using Gaia...
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    A How do I define Open cluster membership?

    I'm doing a study using Gaia DR2 data, and in particular trying to analyse the photometric data for groups of stars which some authors believe to be loosely gravitationally bound in open clusters. I am trying to find out a reasonable way of establishing cluster membership, before analysing the...
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    Direction of propagation of an EM wave

    Ah, well I can see that the term in the exponential brackets must be constant in order for there to be no attenuation over time, so I imagine that y must increase as t increases...hence propagation in the y-direction. So far so good. So I guess at z=0 the expression is zero, and this indicates...
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    Direction of propagation of an EM wave

    I'm reading my course book on ELectromagnetism and it is talking about a wave moving in the y-z plane but with polarisation in the x-direction, and it says that the equation $$\mathbf{E}=E_0 2i sin(k_0 z\ cos \theta) exp[i(k_0 y\ sin \theta - \omega t)]\mathbf{e}_x$$ Shows that there is no...
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    Object falling from rest, solution of differential equation

    Thanks a million. It's been a while since I've done this, and I seemed to be making a mistake at each stage...
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    Object falling from rest, solution of differential equation

    Ok, I take your point...I just thought maybe people had thought I was referring to v(x) which also gets used in kinematics quite a lot. The problem I had was really how to separate the variables. Anyway, having spent most of the day on this (!) I can now see how the separation of variables is...
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    Object falling from rest, solution of differential equation

    Still not right: I'm getting $$x(t)=\left(C+Dt\right)e^{-\frac{k}{m}t}+\frac{mg}{k}t$$ So taking out C, as I'll define the object to be at the origin at t=0, and differentiating: $$v(t)=e^{-\frac{k}{m}t}D\left(1-\frac{k}{m}t\right)+\frac{mg}{k}$$ Using initial conditions (object at rest)...
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    Object falling from rest, solution of differential equation

    Ahhh, thanks. That makes sense now. So $$\lambda_1=\lambda_2=-\frac{k}{m}$$ so $$x(t)=\left(C+Dt\right)e^{-\frac{k}{m}t}$$ If I include the fact that I made a mistake with the particular integral (should have been mg/k, thanks Lawrence) I might be nearly there. EDIT: I put the wrong...
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    Object falling from rest, solution of differential equation

    Thanks for both of these replies. Firstly, to Marioeden: Yes, that's the separation of variables technique that I'm familiar with. I usually look to separate the variables so that they are on either sides of the equation, then perform the integration using the chain rule as you mentioned...
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    Object falling from rest, solution of differential equation

    Homework Statement An object is falling from rest with air resistance modelled by $$kv_x$$. Where v_x is the object's velocity in the x-direction (downwards). Find an expression for the speed of the object as a function of time. Homework Equations $$mg-kv_x=m\frac{dv_x}{dt}$$ The...
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