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When an object with mass \(m\) falls under gravity, its displacement \(x\) is governed by the equation of motion:$$ m\ddot{x} = -mg - k(\dot{x})^2 $$where \(g \approx 9.8\) is the acceleration due to gravity and \(k\) is a positive constant representing air resistance.
Use the substitution \(v = \dot{x}\) to write this as a coupled system of differential equations involving \(x\) and \(v\).
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