Midterm2PracticeProblems.pdf - MATH 32A Spring 2018 Midterm Exam 2 Practice Problems 1 Show that for an arc-length parameterisation and some constant k

Midterm2PracticeProblems.pdf - MATH 32A Spring 2018 Midterm...

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MATH 32A Spring 2018, Midterm Exam 2 Practice Problems 1. Show that, for an arc-length parameterisation and some constant k , d B d s = k N . Recall B ( s ) = T ( s ) × N ( s ), and/or try differentiating B · B . 2. Let κ ( t ) be the curvature of a path r ( t ) = ( x ( t ) , y ( t ) , z ( t )). Show that the scaled path r 1 ( t ) = ( λx ( t ) , λy ( t ) , λz ( t )) has curvature κ ( t ) . Use κ ( t ) = r 0 ( t ) × r 00 ( t ) || r 0 ( t ) || 3 . 3. A particle follows the path r ( t ) for 0 t T , starting from the origin, r (0) = 0. The quantity ¯ v = 1 T Z T 0 r 0 ( t ) d t is called the average velocity. Suppose the ¯ v = 0 . Where is the particle at time T ? Is the particles average speed ( 1 T R T 0 v d t ) also equal to zero? 4. Find the path of an object starting at position r 0 = (0 , 0) with initial velocity v 0 = (1 , 3), with constant acceleration a = ( - 1 , - 1). Find the position of the object at time t = 2. 5. Decompose the acceleration a ( t ) at t = 0 into components parallel and perpendicular to r 0 (0) for an object following the path r ( t
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