Variable Acceleration, Arc Length and Tangent Vector - Vector Geometry | MATH 1224, Study notes of Vector Analysis

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Unit 10

Saturday, November 13, 2010 7:04 PM

Unit 10 Saturday, November 13, 2010 7:04 PM Math 1224 Vector Geometry. UNIT 10: Variable Acceleration, Arc Length, and Tangent Vector Reminders: This is our last unit of new material. However, you have three deadlines left in order to complete the course. Please do not forget them: iors Nov |7 Quiz 10: due by Review for Test 3: Nov |&* 14 Test 3: due by DEC. LC 4.00Pm) Review for Final Exam: Dec a+ 3 oS OOP mn Final Examy sy fagig dueby Dec 1% > (check the ontine schedule for specific hours) Dec § Review: The Unit Tangent Vector In the last unit, we discussed the unit tangent vector. The unit tangent vector T(t) to a trajectory 7(t) at a point determined by f = fo is a unit vector whose direction is the same as the direction the trajectory is moving. Since the velocity vector points in the direction the trajectory is moving, we can find T(t) by first finding the velocity vector, and then scaling it to a length of 1 unit. ‘The unit normal vector N(t), which you saw in recitation last week, is orthogonal to the unit tangent vector, and can be found using the formula: 1 4 N(t) = ——T'(t). IP'O)|) Tangential Component of Acceleration Let s(t) bea trajectory with acceleration (t). The tangential component of acceleration, ar, is the scalar projection of d onto the unit tangent vector T. (We covered vector and scalar projections in Unit 5.) Normal Component of Acceleration ‘Ihe normal component of acceleration, ay, is the scalar projection of d onto the unit normal vector N. We can use the formula ay = d+ N to calculate this component. Lucture Notes Page |