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This is a standard projectile motion problem with the complication that the projectile collides elastically with vertical walls. Assuming that the only force exerted on the projectile by the walls is in a direction which is normal to the walls then: the vertical velocity does not change, and the horizontal velocity reverses direction but the magnitude ...

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Of course, you can do it with Matlab, Mupad, Maple, Mathematica or even the Smart Math Calculator. Use this method: First define your variables with your units of choice, then tell the programm what the conversion factors from the given units to the target units are, for example, if you have km/h and need m/sec define 1km as 1000m and 1sec as h as 60² ...

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With forward Euler, you're simply out of luck. Because of the way the scheme is constructed, you always make an error in momentum in the same direction, which then compounds, leading to exponential behavior. You need a symplectic integrator, the simplest being leapfrog, or any other Verlet integrator. These still won't conserve momentum on a timestep to ...

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The proof is very simple and comes right form the definition. let $R(t)$ be a rotation matrix as a function of time. $R$ is an orthogonal matrix so its inverse is equal to its transpose: $I = R(t)R^T(t).$ ($I$ is the identity matrix) The time derivative of the above equation is \$0 = \frac{d[R(t)R^T(t)]}{dt} = ...

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