479 Horizontal Motion: The horizontal component of velocity ... - Wuala

More documents

Recommendations

Info

91962_05_R1_p0479-0512 6/5/09 3:56 PM Page 510 © 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. R1–47. A projectile of mass m is fired into a liquid at an y angle u 0 with an initial velocity v 0 as shown. If the liquid develops a friction or drag resistance on the projectile which is proportional to its velocity, i.e., F = -kv, where k is a constant, determine the x and y components of its position at any instant. Also, what is the maximum distance x max that it travels? Equation of Motion: Applying Eq. 13–7, we have O v 0 0 u x : + ©F x = ma x ; -ky x = ma x a x = - k m y x [1] + c ©F y = ma y ; -mg - ky y = ma y a y = -g - k m y y [2] However, , , and a y = d2 y a y y = dy x = d2 x y x = dx . Substituting these values into dt dt 2 dt dt 2 Eqs. [1] and [2], we have d 2 x [3] dt 2 + k dx m dt = 0 d 2 y [4] dt 2 + k dy m dt = -g The solution for the differential equation, Eq. [3], is in the form of Thus, x = C 1 e - k m t + C 2 x # C 1 k = - k m e- m t [5] [6] However, at t = 0, x = 0 and x # = y 0 cos u 0 . Substituting these values into Eq. [5] m and [6], one can obtain C and C 2 = m 1 = - . Substituting C k y 0 cos u k y 0 cos u 0 0 1 and C 2 into Eq. [5] and rearrange yields x = m k y 0 cos u 0 A1 - e - k m t B Ans. When t : q, e - k m t : 0 and x = x max . Then, x max = m k y 0 cos u 0 The solution for the differential equation. Eq. [4], is in the form of y = C 3 e - k m t + C 4 - mg k t Ans. [7] Thus, y # C 3 k = - k m e- m t - mg k [8] However, at t = 0, y = 0 and y # = y 0 sin u 0 . Substitute these values into Eq. [7] and [8], one can obtain C m and C 4 = m . k ay 0 sin u 0 + mg 3 = - k ay 0 sin u 0 + mg k b k b Substitute C 3 and C 4 into Eq. [7] and rearrange yields y = m k ¢y 0 sin u 0 + mg k ≤ a1 - e- k m t b - mg k t Ans. 510
91962_05_R1_p0479-0512 6/5/09 3:56 PM Page 511 © 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. *R1–48. The position of particles A and B are r and r B = 53(t 2 A = 53ti + 9t(2 - t)j6 m - 2t + 2)i + 3(t - 2)j6 m, respectively, where t is in seconds. Determine the point where the particles collide and their speeds just before the collision. How long does it take before the collision occurs? When collision occurs, r A = r B . 3t = 3(t 2 - 2t + 2) t 2 - 3t + 2 = 0 t = 1 s, t = 2 s Also, 9t(2 - t) = 3(t - 2) 3t 2 - 5t - 2 = 0 The positive root is t = 2 s Thus, t = 2 s Ans. x = 3(2) = 6 m y = 9(2)(2 - 2) = 0 Hence, (6 m, 0) Ans. v A = dr A dt = 3i + (18 - 18t)j v A | t = 2 = {3i - 18j} m>s v A = 2(3) 2 + (-18) 2 = 18.2 m>s Ans. v B = dr B dt = 3(2t - 2)i + 3j v B | t = 2 = {6i + 3j} m>s v B = 2(6) 2 + (3) 2 = 6.71 m>s Ans. 511
Page 1 and 2: 91962_05_R1_p0479-0512 6/5/09 3:53
Page 3 and 4: 91962_05_R1_p0479-0512 6/5/09 3:53
Page 5 and 6: 91962_05_R1_p0479-0512 6/5/09 3:53
Page 7 and 8: 91962_05_R1_p0479-0512 6/5/09 3:53
Page 9 and 10: 91962_05_R1_p0479-0512 6/5/09 3:53
Page 11 and 12: 91962_05_R1_p0479-0512 6/5/09 3:53
Page 13 and 14: 91962_05_R1_p0479-0512 6/5/09 3:53
Page 15 and 16: 91962_05_R1_p0479-0512 6/5/09 3:53
Page 17 and 18: 91962_05_R1_p0479-0512 6/5/09 3:54
Page 19 and 20: 91962_05_R1_p0479-0512 6/5/09 3:54
Page 21 and 22: 91962_05_R1_p0479-0512 6/5/09 3:54
Page 23 and 24: 91962_05_R1_p0479-0512 6/5/09 3:54
Page 25 and 26: 91962_05_R1_p0479-0512 6/5/09 3:55
Page 27 and 28: 91962_05_R1_p0479-0512 6/5/09 3:55
Page 29 and 30: 91962_05_R1_p0479-0512 6/5/09 3:55
Page 31: 91962_05_R1_p0479-0512 6/5/09 3:56

479 Horizontal Motion: The horizontal component of velocity ... - Wuala

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?