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A particle moves along x-axis and is undergoing deceleration which is proportional to the cube of its instantaneous speed v. The initial speed is v0 and after describing a path length x = x0, the speed is reduced to v_0/2. Then (a) when x = 5x0, the speed is v_0/16 (b) velocity of the particle is given by v = v_0 e^(-x/x_0 ) (c) velocity of the particle is given by v = v_0/(1+x/x_0 ) (d) when x = 3x0, the speed is v_0/5
(c) Given a = –v3 a = –kv3 where k is a constant of proportionality –v dv/dx = kv3 Integrating ∫¦?v^(-2) dv?=-∫¦?k dx? –1/v = –kx + C [C is constant of integration] When x = 0, v = v0 (-1)/v_0 = C 1/v=-kx-1/v_0 1/v-1/v_0 = kx … (1) Also when x = x0, v = v_0/2 2/v_0 -1/v_0 = kx0 k = 1/(v_0 x_0 ) From (1), we get v = v_0/(1+x/x_0 ) When x = 3x0 v = v_0/4
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