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Unformatted text preview: homework 03 ASH, BEN Due: Feb 6 2008, 11:00 pm 1 Question 1, chap 3, sect 99. part 1 of 2 10 points A ship cruises forward at v s = 6 m / s rel ative to the water. On deck, a man walks diagonally toward the bow such that his path forms an angle = 19 with a line perpen dicular to the boats direction of motion. He walks at v m = 2 m / s relative to the boat. Draw the vectors to scale on a graph to determine the answer. v s v m At what speed does he walk relative to the water? Correct answer: 6 . 91474 m / s (tolerance 5 %). Explanation: v s v v m Scale: 1 m Let : v s = 6 m / s v m = 2 m / s v = 6 . 91474 m / s = 71 = 55 . 1286 = 161 , and = 15 . 8714 . When you complete the parallelogram, the resultant velocity v with respect to the water is the side of the triangle opposite the obtuse angle, which has a measure of = 90 + . Let vectorv s be the velocity of the ship, vectorv m be the velocity of the man, and vectorv be the resultant velocity of the man relative to the water (Earth). By the law of cosines v 2 = v 2 m + v 2 s 2 v m v s cos v = bracketleftbig v 2 m + v 2 s 2 v m v s cos bracketrightbig 1 / 2 = bracketleftbig (2 m / s) 2 + (6 m / s) 2 2(2 m / s)(6 m / s) cos(109 )] 1 / 2 = bracketleftbig (2 m / s) 2 + (6 m / s) 2 +(7 . 81363 m 2 / s 2 ) bracketrightbig 1 / 2 = 6 . 91474 m / s . Alternatively: We can analyze the vector addition using the components of the vectors. Note: vectorv = vectorv s + vectorv m , or v x = v s + v m sin = (6 m / s) + (2 m / s) sin(19 ) = 6 . 65114 m / s and v y = v m cos = (2 m / s) cos(19 ) = 1 . 89104 m / s Hence the speed of the man with respect to the water is v = radicalBig v 2 x + v 2 y = radicalBig (6 . 65114 m / s) 2 + (1 . 89104 m / s) 2 = 6 . 91474 m / s . Question 2, chap 3, sect 99. part 2 of 2 10 points At what angle to his intended path does the man walk with respect to the water? Correct answer: 55 . 1286 (tolerance 5 %). Explanation: The law of sines can be used to compute the requested angle , which is the angle opposite the ships path and velocity. sin v s = sin v homework 03 ASH, BEN Due: Feb 6 2008, 11:00 pm 2 sin = v s v sin = arcsin bracketleftBig v s v sin bracketrightBig = arcsin bracketleftbigg 6 m / s 6 . 91474 m / s sin(109 ) bracketrightbigg = 55 . 1286 . Alternate Solution: Using vector compo nents from Part 1, we have tan = v y v x = arctan parenleftbigg v y v x parenrightbigg = arctan parenleftbigg 1 . 89104 m / s 6 . 65114 m / s parenrightbigg = 15 . 8714 . Therefore the angle between vectorv m and vectorv is = 90 = 90 (19 ) (15 . 8714 ) = 55 . 1286 ....
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This note was uploaded on 09/29/2008 for the course PHY 303 taught by Professor Erskine/tsoi during the Spring '08 term at University of Texas at Austin.
 Spring '08
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