Motion Along a Line: Vectors

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Presentation transcript:

Motion Along a Line: Vectors March 30th, 2017

Parametric Equations-Notation If a smooth curve C is given by x=f(t) and y=g(t), then represents the slope of C at the point (x, y), or the slope of the path of a particle traveling along C, or the rate of change of C with respect to x. is the rate at which the x-coordinate is changing with respect to t (velocity of the particle in the horizontal direction). is the rate at which the y-coordinate is changing with respect to t (velocity of the particle in the vertical direction).

Vectors & Calculus is the position vector at any time t. is the velocity vector at any time t. is the acceleration vector at any time t. is the speed of the particle or magnitude (length) of the velocity vector. is the arc length of the curve C from t=a to t=b, or the distance traveled by the particle from time [a,b].

Ex. 1: A particle moves in the xy-plane so that at any time t, the position of the particle is given by Find the velocity vector when t=1. (b) Find the acceleration vector when t=2.

Ex. 2: A particle moves in the xy-plane so that at any time t, , the position of the particle is given by . Find the magnitude of the velocity vector when t=1.

Ex. 3: A particle moves in the xy-plane so that and , where . The path of the particle intersects the x-axis twice. Find the distance traveled by the particle between the two x-intercepts. (You will need to use a calculator).

Day 2: Ex. 4: An object moving along a curve in the xy-plane has position ((x(t), y(t)) at time t with . At time t=2, the object is at the position (1, 4). Find the acceleration vector for the particle at t=2. Write the equation of the tangent line to the curve at the point where t=2. Find the speed of the vector at t=2. Find the position of the particle at time t=1.

*The examples and definitions in this presentation are courtesy of Nancy Stephenson of Clements High School in Sugar Land, Texas, from the “Vectors in AP Calculus”.