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Analytical Mech Homework Solutions 20

Analytical Mech Homework Solutions 20 - Integrating 11A = t...

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Integrating 1 1 A t u u m α = D and substituting v e u α = (a) 1 ln 1 v A e t m α v v α α = + D D (b) t T @ 0 v = = ln 1 v A v e m α T α α = + D D 1 v v A e e m α α T α = + D D 1 v m T e A α α = D (c) v dv A e dx m v v α = = − v vdv A dx e m α = − again, let u v e α = du udv α = or du dv u α = 1 ln v u α = 1 ln du u A u dx u m α α = − Integrating and solving ( ) 2 1 1 v m x v A α α α = + D D e 2.20 ( ) d mv F mv vm dt = = + = mg but 3 4 3 m r ρ π = D 2 1 m r ρ π = v so (1) 3 2 2 2 1 4 4 3 3 r v r v r g πρ πρ π πρ + = = D D 3 4 3 Now 3 1 10 ρ ρ D so, second term is negligible-small hence
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