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Notes for 11.3 11.4 - Calculus III 11.3 Velocity and...

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Calculus III - 11.3 – Velocity and Acceleration Definition of Velocity and Acceleration If x and y are twice-differentiable functions of t , and r is a vector-valued function given by r ( t) = x(t) I + y(t) j , then the velocity vector, acceleration vector and speed at time t are as follows: Velocity = v ( t) = r (t) = x’(t) i + y’(t) j Acceleration = a (t) = r ’’ (t) = x’’(t) i + y’’(t) j Speed = [ ] [ ] 2 2 ) ( ) ( ) ( ' ) ( t y t x t r t v + = = For motion along a space curve, the definitions are extended in the normal way. Examples: #6 (What do you notice about the velocity and acceleration vectors at the point?) #12
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11.3.2 Position Function for a Projectile Neglecting air resistance, the path of a projectile launched from an initial height h with initial speed v 0 and angle of elevation θ is described by the vector function ( 29 ( 29 j gt t v h i t v t r - + + = 2 0 0 2 1 sin cos ) ( θ θ where g is the gravitational constant. #28 #36
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Calculus III – Section 11.4 – Tangent Vectors and Normal Vectors Let C be a smooth curve represented by r
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