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Vector Valued Functions

Vector Valued Functions. Velocity and Acceleration. Written by Judith McKaig Assistant Professor of Mathematics Tidewater Community College Norfolk, Virginia. Definitions of Velocity and Acceleration:

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Vector Valued Functions

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  1. Vector Valued Functions Velocity and Acceleration Written by Judith McKaig Assistant Professor of Mathematics Tidewater Community College Norfolk, Virginia

  2. Definitions 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: The definitions are similar for space functions of the form: r(t) = x(t)i + y(t)j + z(t)k

  3. Example 1: The position vector • describes the path of an object moving in the xy-plane. • Sketch a graph of the path. • Find the velocity, speed, and acceleration of the object at any time, t. • Find and sketch the velocity and acceleration vectors at t = 2 The curve can then be represented by the equation with the orientation as shown in the graph. Solution: a. To help sketch the graph of the path, write the following parametric equations:

  4. b. So the following vector valued functions represent velocity and acceleration and the scalar for speed:v(t) = i + 2tja(t) = 2j c. At t = 2, plug into the equations above to get:the velocity vector v(2) = i + 4j, the acceleration vector a(2) = 2j To sketch the graph of the velocity vector, start at the initial point (2,4) and move right 1 and up 4 to the terminal point (3,8). Sketch the acceleration similarly.

  5. Example 2: The position vector • describes the path of an object moving in the xy-plane. • Sketch a graph of the path. • Find the velocity, speed, and acceleration of the object at any time, t. • Find and sketch the velocity and acceleration vectors at (3,0) Solution: a. To help sketch the graph of the path, write the following parametric equations: Since , the curve can be represented by the equation which is an ellipse with the orientation as shown in the graph.

  6. b. By differentiating each component of the vector, you can find the following vector valued functions which represent velocity and acceleration. You can use the formula to find the scalar for speed:v(t) = -3sinti + 2costja(t) = -3costi-2sintj r(t) = 3costi + 2sintj c. The point (3,0) corresponds to t = 0. You can find this by solving: 3cos t = 3cos t = 1t = 0 At t = 0, the velocity vector is given by v(0) = 2j, and the acceleration vector is given by a(0) = -3i

  7. Solution: Recall, you are given r(t) in component form. It can be written in standard form as: The velocity and acceleration can be found by differentiation: The speed is found using the formula and simplifying: Example 3: The position vector r describes the path of an object moving in space. Find the velocity, acceleration and speed of the object.

  8. For comments on this presentation you may email the author Professor Judy Gill at jgill@tcc.edu or the publisher of the VML, Dr. Julia Arnold at jarnold@tcc.edu.

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