PS1-6 - V 3 i We can now determine the value of C by using...

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Problem Set 1: Problem 6. Problem: The acceleration of a particle is a = C/v ,whe re C is a constant. At time t =0 the particle is located at x =0 and its velocity is v = V . Also, when its position is x = f the particle’s velocity is v =2 V . (a) Determine the constant C . (b ) A twha tt ime τ is the particle’s velocity v =3 V ? Solution: Since we know the acceleration as a function of velocity, the obvious first step in solving this problem is to begin with the differential equation relating acceleration and velocity, viz., a = v dv dx = adx = vdv (a) For the given acceleration, we have C v dx = vdv = dx = 1 C v 2 dv Then, integrating, we have 8 x o dx = 1 C 8 v V v 2 dv = x = 1 3 C v 3 e e e e v = v v = V Thus, the particle’s position as a function of its velocity is x = 1 3
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Unformatted text preview: V 3 i We can now determine the value of C by using the fact that x = f when v = 2 V . Hence, f = 1 3 C D 8 V 3 V 3 i = 7 3 C V 3 Finally, solving for C yields C = 7 3 V 3 f (b) For this part of the problem, we must use the definition of acceleration, i.e., a = dv dt = dv = a dt = C v dt Hence, we can determine the time from dt = 1 C v dv = 3 7 f V 3 v dv Integration shows that = 8 dt = 3 7 f V 3 8 3 V V v dv = 3 14 f V 3 v 2 e e e e v =3 V v = V = 3 14 f V 3 D 9 V 2 V 2 i Therefore, the time at which the particles velocity is v = 3 V is = 3 14 f V 3 D 8 V 2 i = 12 7 f V...
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