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lec_week3

# lec_week3 - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 200 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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18.02 Lecture 6. Tue, Sept 18, 2007 Handouts: Practice exams 1A and 1B. Velocity and acceleration. Last time: position vector r ( t ) = x ( t ı + y ( t j [+ z ( t ) k ˆ ] . E.g., cycloid: r ( t ) = t sin t, 1 cos t . d� r dx dy Velocity vector: �v ( t ) = dt = dt , dt . E.g., cycloid: �v ( t ) = 1 cos t, sin t . (at t = 0, �v = 0: translation and rotation motions cancel out, while at t = π they add up and �v = 2 , 0 ). Speed (scalar): | �v | . E.g., cycloid: | �v | = (1 cos t ) 2 + sin 2 t = 2 2 cos t . (smallest at t = 0 , 2 π, ... , largest at t = π ). d�v Acceleration: �a ( t ) = . E.g., cycloid: �a ( t ) = sin t, cos t (at t = 0 �a = 0 , 1 is vertical). dt Remark: the speed is | d� r | , which is NOT the same as d | r | ! dt dt ds Arclength, unit tangent vector. s = distance travelled along trajectory.
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