MATHEMATICS 317 T2 Assignment1(2)

# MATHEMATICS 317 T2 Assignment1(2) - MATHEMATICS 317...

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MATHEMATICS 317 Assignment #1 due at the beginning of class on Wednesday, January 13 th 1. A particle moves in the xy -plane with velocity . ) / 2 ( / 2 j i r r v x x dt d - = = = & (a) Find the position of the particle, i.e., r ( t ) at time t , if r (0) = i + j . (b) Find the acceleration of the particle. 2. Suppose j i r ) cos 1 ( ) sin ( ) ( t t t t - - = is the position of a particle at time t . (a) Find the times when the particle has zero velocity. (b) Find the times when the particle has its greatest speed. 3. If , / ) ( dt d t a a b × = show that . / / ) ( 2 2 dt d dt t d a a b × = 4. Simplify .. 2 2 × dt d dt d dt d r r r 5. (a) Use properties 5 and 6 on p. , i.e., , ) ( ) ( ) ( , ) ( ) ( c b a b c a c b a c b a c b a - = × × × = × to show that ). )( ( ) )( ( ) ( ) ( d a c b d b c a d c b a - = × × (b) Consider two particles located at positions ) ( 1 t r and ), ( 2 t r respectively, with corresponding velocities given respectively by ) ( 1 t v and ). ( 2 t v Consider the triangle formed by the origin and the position vectors of the two particles. Show that the rate of change of the area A ( t ) of this triangle is given by the expression

Unformatted text preview: . | | 2 )] ( ) )[( ( ) ( | | ) ( | | 1 2 1 2 2 1 1 2 2 2 2 1 1 1 2 2 r r r v r v r r r v r r v r × ⋅ + ⋅ ⋅-⋅ + ⋅ = dt dA 6. Find the unit tangent vector at each point of the curve . ) 6 1 ( ) 3 2 ( ) 2 1 ( ) ( k j i r t t t t + +-+ + = What does your answer imply about the curve? Find the arc length between the points where t = 0 and t = 1. 7. Consider the curve given by . 2 ) ( k j i r t e e t t t + + =-(a) Show that this curve lies on the surfaces of the cylinders xy = 1 and . 2 / z e x = (b) Find the unit tangent vector at each point of the curve. (c) Find the arc length of the curve from t = 0 to t = 1. 8. Consider the curve given by . sin cos ) ( k j i r t t t t t t + + = (a) Show that this curve lies on the surface of a cone. (b) Find the arc length of the curve from t = 0 to t = π ....
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