2006 Final

2006 Final - FACULTY OF SCIENCE FINAL EXAMINATION PHYS 230...

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FACULTY OF SCIENCE FINAL EXAMINATION PHYS 230 DYNAMICS OF SIMPLE SYSTEMS Examiner: Prof. R. Harris Friday December 8th, 2006 Associate Examiner: Prof. N. de Takacsy 2:00 - 5:00 pm INSTRUCTIONS: Answer all questions in the booklet provided. There are 5 questions at 10 points each. Textbooks and/or notes are not permitted. However, calculators are permitted, and formulae are provided on the last page. Dictionaries, if required, are also permitted. You may keep the question paper. This exam comprises 5 pages, with questions on pages numbered 1 through 3.
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PHYS 230 1 1. This problem was provided as an “extra” before the MidTerm. Two boats are playing on a lake – or, rather, their drivers are playing. One boat, “Albert”, is traveling due East (left to right), and the second, “Brigitte”, starts 500 metres to the North, and tries to meet up with the first. - ? w w Albert Brigitte There are three scenarios. In all three, Albert always travels due East. At the beginning of each scenario, Albert is always travelling at 4 m/s, and Brigitte at 5 m/s. Take the origin of coordinates to be the original position of Albert. In each scenario, Brigitte always attempts to travel directly towards Albert, so as to meet in the shortest possible time. . You are advised to solve all three scenarios by working in Brigitte’s frame of reference : this should make the solution(s) easier. (a) Albert has constant velocity 4 m/s due East, Brigitte has constant velocity 5 m/s. Show that, in order that they meet, Brigitte must travel at an angle θ = arccos (4 / 5) South of East. In Brigitte’s frame, Albert will always be travelling towards her. This means that she will always match the component of his velocity that is perpendicular to the line from her to him. Therefore, the component of her velocity to the East must be 4 m/s. And, conse- quently, the component of her velocity to the South must be 3 m/s, so that she travels at an angle θ = arccos (4 / 5) South of East. Z Z Z Z Z~ ? - 4 5 3 What is the time and place of the meeting? In her frame, he has to travel a distance of 500 m with a velocity of 3 m/s. This takes 500/3 seconds. The place of meeting is 500 / 3 × 4 ± 667 metres to the East of Albert’s initial position. (b) Albert now accelerates at 0.01 m/s 2 , starting at 4 m/s due East, and in response, starting at 5 m/s, in the direction θ = arccos (4 / 5) South of East, Brigitte accelerates at 0.0125 m/s 2 . What is the time and place of the meeting?
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2006 Final - FACULTY OF SCIENCE FINAL EXAMINATION PHYS 230...

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