ism_chapter_03 - Chapter 3 Motion in Two and Three...

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123 Chapter 3 Motion in Two and Three Dimensions Conceptual Problems *1 Determine the Concept The distance traveled along a path can be represented as a sequence of displacements. Suppose we take a trip along some path and consider the trip as a sequence of many very small displacements. The net displacement is the vector sum of the very small displacements, and the total distance traveled is the sum of the magnitudes of the very small displacements. That is, total distance = N N 1, 3 , 2 2 , 1 1 , 0 ... + + + + r r r r r r r r where N is the number of very small displacements. (For this to be exactly true we have to take the limit as N goes to infinity and each displacement magnitude goes to zero.) Now, using the shortest distance between two points is a straight line, we have N N N 1, 3 , 2 2 , 1 1 , 0 , 0 ... + + + + r r r r r r r r r r , where N , 0 r r is the magnitude of the net displacement. Hence, we have shown that the magnitude of the displacement of a particle is less than or equal to the distance it travels along its path. 2 Determine the Concept The displacement of an object is its final position vector minus its initial position vector ( i f r r r r r r = ). The displacement can be less but never more than the distance traveled. Suppose the path is one complete trip around the earth at the equator. Then, the displacement is 0 but the distance traveled is 2 π R e .
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Chapter 3 124 3 Determine the Concept The important distinction here is that average velocity is being requested, as opposed to average speed . The average velocity is defined as the displacement divided by the elapsed time. 0 0 av = = = t t r v r r circuit. complete each of end the at zero always is velocity average the s, car travel race fast the how matter no that see we Thus zero. is track the around any trip for nt displaceme The What is the correct answer if we were asked for average speed ? The average speed is defined as the distance traveled divided by the elapsed time. t v distance total av zero. not is average the and zero an greater th be will traveled distance total the any track, of circuit complete one For 4 False. Vectors are quantities with magnitude and direction that can be added and subtracted like displacements. Consider two vectors that are equal in magnitude and oppositely directed. Their sum is zero, showing by counterexample that the statement is false. 5 Determine the Concept We can answer this question by expressing the relationship between the magnitude of vector A r and its component A S and then using properties of the cosine function. Express A S in terms of A and θ : A S = A cos Take the absolute value of both sides of this expression: A S = A cos = A cos and cos = A A S
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Motion in One and Two Dimensions 125 Using the fact that 0 < cos θ ⎜≤ 1, substitute for cos to obtain: 1 0 S < A A or A A < S 0 vector. the of magnitude the to equal or than less be must vector a of component a of magnitude The No.
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This note was uploaded on 08/18/2010 for the course IMT 12253 taught by Professor Mikael during the Spring '10 term at École Normale Supérieure.

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ism_chapter_03 - Chapter 3 Motion in Two and Three...

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