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Section_1_Vector_algebra - Section 1 Vector algebra Problem...

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Section 1 Vector algebra Problem 1 The structure of a methane (CH 4 ) molecule is such that the carbon atom is positioned at the « center » of the four hydrogen atoms. The four hydrogen atoms are at positions A (0, 0, 0), B (1, 0, 1), C (1, 1, 0), D (0, 1, 1). a) Find the carbon atom’s position ( E ). b) Find the angle separating segments EA from EB. Note : the carbon atom is at the « center » if the sum of the vectors joining it to the hydrogen atoms is zero. Solution a) E will be at the center of the molecule if 2 1 , 2 1 , 2 1 4 1 4 0 0 E D C B A E E D C B A E D E C E B E A ED EC EB EA b) 5 . 109 3 1 cos cos 4 3 4 1 cos 4 3 4 3 2 1 , 2 1 , 2 1 2 1 , 2 1 , 2 1 cos EB EA EB EA
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Problem 2 Ball # 1 (0.05 kg) is originally at rest. The motion of ball 2 (0.10 kg) is described by the velocity vector 4 , 3 2 i v . The two balls are involved in a collision, following which ball 2’s velocity vector is 6 . 3 , 4 . 2 2 f v . If momentum is conserved throughout this experience, find a) 1( ) f v v , the vector describing ball 1’s velocity after the collision ; b) , the angle with which the balls separate. Note : when momentum is conserved k k k i k f m v m v v v Bonus : is this collision elastic ? Solution a) 8 . 0 , 2 . 1 6 . 3 , 4 . 2 2 4 , 3 2 6 . 3 , 4 . 2 10 . 0 4 , 3 10 . 0 05 . 0 6 . 3 , 4 . 2 10 . 0 05 . 0 4 , 3 10 . 0 0 , 0 05 . 0 1 1 1 1 2 2 1 1 2 2 1 1 f f f f f f i i v v v v v m v m v m v m b)  .2 0 5 cos cos 00 . 9 5.76 cos 33 . 4 2.08 6 . 3 , 4 . 2 8 . 0 , 2 . 1 cos 2 1 2 1 f f f f v v v v c) The collision is not elastic as (Kinetic) energy is not conserved since 25 . 1 i K whereas 982 . 0 f K
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Problem 3 Find the value(s) of k such that the vector 2 1 , , 2 1 k v will be a) a unit vector b) parallel to 2 , 5 , 2 u c) perpendicular to the vector 2 2 , 3 , 8 k w Solution a) We wish to find k such that 1 1 2 v v 2 1 2 1 1 4 1 4 1 1 2 1 2 1 2 2 2 2 2 k k k k b) If v u // , then v a u 4 5 4 5 5 4 2 1 2 2 1 , , 2 1 2 , 5 , 2 k k k a a a k a c) If w u , then 0 w u  1 , 4 0 1 4 0 4 3 0 3 4 0 2 , 3 , 8 2 1 , , 2 1 2 2 2 k k k k k k k k k k
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Problem 4 Let A be a point whose coordinates are ( 2 , 3 ).
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