Motion_and_Curvature9-2&3

Find the speed and position at any time instant t at

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Unformatted text preview: j. Find the speed and position at any time instant t. At what angle should the airplane with speed 180 mph release the supply pack to hit the target? Gravitional acceleration : a = − gj 5 6 Examples Engineering and Physics Examples Example Graph the trace of a movingparticle with position vector Movement of planets around sun r ˆ r (t ) = 2 cos t ˆ + 2 sin t ˆ + 3 k i j Mm u, r =|| r ||, u = r / r r2 d dP F = ma = (mv ) = dt dt F = −k Solution: x 2 + y 2 = 4, z = 3 Circle in a Plane Movement of satellites around ground 7 8 2 1/2/2011 Example: Graph the particle position of a moving particle given by t 0 π/2 x 2 0 y 0 2 z 0 π/2 π 3π / 2 2π 5π / 2 3π 7π / 2 4π 9π / 2 −2 0 2 0 −2 0 2 0 0 −2 0 2 0 −2 0 2 r r (t ) = 2 cos t ˆ + 2 sin t ˆ + t k i jˆ Length of a Space Curve (s): If r(t) is a smooth function, then the length of the smooth curve traced by r(t) is given by x 2 + y 2 = 4; z = t π 3π / 2 2π 5π / 2 3π 7π / 2 4π 9π / 2 Circular Helix r with r (t ) =< f (t ), g (t ), h(t ) >, then b s = ∫ || r ' (t ) ||dt t=a t=b a x 2 + y 2 = 4, z = 3 b = ∫ [ f ' (t )]2 + [ g ' (t )]2 + [h' (t )]2 dt a In mechanics when r(t) is the position ve...
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This note was uploaded on 06/30/2011 for the course ENGR 233 taught by Professor S.samuelli during the Winter '10 term at Concordia Canada.

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