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    Geometric Optics - Magnification

    Homework Statement A concave mirror forms an image on a screen twice as large as an object. Both object and mirror are then moved such that the new image is 3x the size of the object. If the screen is moved 75cm, how far did the object move? Homework Equations m = image distance / object...
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    Solving second-order ODE with Runge-Kutta 4

    Homework Statement Note: I think there is a typo here but I'm not sure. Is there supposed to be a comma between the delta t/2 and y_n on K2 and K3, and delta t and y_n on K4? Homework Equations See above. The Attempt at a Solution Substituting dy/t = z gives \frac{dz}{dt} = 3z - 2ty -...
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    Polynomial finite fields; ElGamal decryption

    Homework Statement Given some ElGamal private key, and an encrypted message, decrypt it. Homework Equations Public key (F_q, g, b) Private key a such that b=g^a Message m encrypted so that r=g^k, t=mb^k Decrypt: tr^-a = m The Attempt at a Solution My problem is finding r^-a...
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    Legendre polynomials and binomial series

    Homework Statement Where P_n(x) is the nth legendre polynomial, find f(n) such that \int_{0}^{1} P_n(x)dx = f(n) {1/2 \choose k} + g(n) Homework Equations Legendre generating function: (1 - 2xh - h^2)^{-1/2} = \sum_{n = 0}^{\infty} P_n(x)h^n The Attempt at a Solution I'm not sure if that...
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    Hill cipher - two plaintext pairs encrypt to same ciphertext

    Homework Statement Given the following key, find two plaintext pairs that encrypt to the same ciphertext. \begin{pmatrix} 9 & 5 \\ 7 & 3 \\ \end{pmatrix} (mod \ 26) Homework Equations The Attempt at a Solution Let the two plaintext pairs be (a, b) and (c, d). Let the...
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    Rocket momentum

    Homework Statement A rocket of initial mass m_{0} accelerates from rest in vacuum in the absence of gravity. As it uses up fuel, its mass decreases but its speed increases. For what value of m is its momentum p = mv maximum? Homework Equations Tsiolkovsky rocket equation: v(m) = v_e ln...
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    Momentum conservation of asteroid in a dust cloud

    Note: this is one of the suggested practice problems for my second-year classical mechanics course. Homework Statement A spherical asteroid of mass m_{0} and radius R, initially moving at speed v_{0}, encounters a stationary cloud of dust. As the asteroid moves through the cloud, it collects...
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    Determine the electric field on the y axis

    Homework Statement Two 2.04 µC point charges are located on the x axis. One is at x = 1.02 m, and the other is at x = -1.02 m. Determine the electric field on the y axis at y = 0.470 m. Homework Equations E = k*q/r^2 The Attempt at a Solution First I find the distance from one...
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    Time a spring takes to slow a mass

    Homework Statement A mass M is attached to a spring with spring constant K. At the equilibrium point of the spring, the mass has a velocity of V. M = 8.07 kg K = 113 N/m V_o = 0.638 m/s How far does the mass travel until it stops? How long (in seconds) does it take for the mass to travel from...
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    Find horizontal asymptotes of a radical function

    Homework Statement Find the horizontal asymptotes for the following equation: Homework Equations f(x) = \sqrt{x^2+4x}-\sqrt{x^2+x} The Attempt at a Solution First I factored f(x): f(x) = \sqrt{x}\sqrt{x+4}-\sqrt{x+1} Then I conjugated it: f(x) =...
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    NaNo_3 dissolved in NaCl saturated solution

    Homework Statement Explain why sodium chloride will precipitate from a saturated aqueous solution when sodium nitrate is dissolved in this solution. Homework Equations no equations. The Attempt at a Solution The saturated NaCl solution cannot dissolve any more sodium, so as NaNo3 dissolves...
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    Dart trajectory, trig proof.

    A target is suspended on a platform. A dart launcher is placed at ground level and aimed directly at the target along the line of sight (the distance between dart and target can vary infinitely). Assume a bottomless pit below the target. The dart is launched, and regardless of speed, it will hit...
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