Word Count : 466

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The problem involves a parachutist falling under gravity with quadratic air resistance. Let’s solve it step by step. Let: * (M) = mass of parachutist with parachute * (v(t)) = velocity at time (t) after parachute opens * (g = 10~\text{m/s²}) * Air resistance (R = \frac{M v^2}{25}) The equation of motion is: [ M \frac{dv}{dt} = Mg - \frac{M v^2}{25} ] Divide through by (M): [ \frac{dv}{dt} = g - \frac{v^2}{25} = 10 - \frac{v^2}{25} ] This is a separable differential equation: [ \frac{dv}{10 - v^2/25} = dt ] Factor (1/25) from the denominator: [ \frac{dv}{10 - \frac{v^2}{25}} = \frac{dv}{\frac{250 - v^2}{25}} = \frac{25 , dv}{250 - v^2} ] So: [ \frac{25 , dv}{250 - ______ _______ ______ ___ _______ _______ ________ _______ ______.
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