By Chinmony Taraphdar
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Additional resources for The Classical Mechanics
1 and I:8i = J.. A u". m di dj Similarly, d
A) A = Axi + Ay} + A/•. Example 10. If Ans. Now let when (Ax , Ay , A) all are constants for constant vector z A. A-)-k rx ry - ( 2 ~ 2 ~ 2~) Ans. V. 2x z i - xy z j + 3y z k = ~(2x2z) + ~(-xiz) + ~(3lz) = 4xz - 2xyz + 3y2. rx = ry Example 12. (r 3r) rz A. r Ans. V. = 3,-3 + 3,-3 = 6,-3. (V2 rn) Example 13. Prove, Ans. We know that V(rn) = == n(n + l)~ - 2. nr n- 2r. n = (n2 - 2n + 3n) ~-2 = (n2 = n(n + 1) ~-2. Example 14. Prove that fnx (a x r) ds + n) ~-2 = 2aV. s a Where is constant vector ADS.
Hence, if at instant, (t + Ot), the mass of rocket with its remaining fuel be (M - ~M) and velocity be (v - Ov) (obeying momentum conservation) then in that said interval M, ~M mass of exhaust gas leaves from the rocket. ~M = small) Dividing throughout by ~t and taking the limit as M ~ 0, we get dv dM M- = - u dt dt ... (1) Here -ve sign is added on the right hand side to indicate that velocity v increases as mass M decreases. So integrating equation (1). t. time, we get v JVodv = 1M -dM -u Mo M 54 The Classical Mechanics => v = Vo - => v = Vo U lOge( ~: ) -UIOgc(~) ...
The Classical Mechanics by Chinmony Taraphdar