1力と運動.pdfVIP

1 exp 2 d x k x dt2 m x eωt X ω2 x eiωt i x e?iωt A B x Aeiωt + Be?iωt x0 dx iωAeiωt iωBeiωt dt 0 iω A B x0 A + B x0 A B 2 x0 x0 x cosωt + i sin ωt + cosωt i sin ωt 2 2 x0 cos ωt ω √k/m eiωt 1 y kv mg 1: m mg kv dv m mg + kv dt v mg/k dv k dt mg v m k mg k log v t + C k m mg ? k t+C ∴ v e m k eC mg v k mg ? k t v 1 e m k mg m 2 ? k t y t + g 1 e m k k mg/k mg/k v mg/k 2 v mg/k O t 2: y mgt/k O t 3: mkv2 dv 2 m mg + mkv dt v √g/k dv g + kv2 dt dv kdt v2 g/k dv √ √ kdt v g/k v + g/k √ 1 v g/k √ √ log kt + C 2 g/k v + g/k C 0 v √g/k tanh √g/kt 3 √ g/k tanh x ?x e e tanh x ex + e?x t at ?at d e e dt log eat + e?at a eat + e?at m [ kg/mt ?kg/mt ] y log e + e /2 k 2γv dv m kx 2γv dt k/m b,2γ/m a 2 d x dx + a + bx 0 dt2 dt x est est 0 2 st st st s e + sae + be 0 s2 + as + b est 0 ∴ s2 + sa + b 0 s γ √ γ s k m m m γ/m √k/m ω ω s γ √ γ 2 2 γ √ γ 2 2 x A exp + ω t + B exp ω t m m m m γ/m ω s 4 x x 0 O t 4: γ/m ω x x 0 O t t 5: γ/m ω 5