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

3. 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

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

5. 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. So the following vector valued functions represent velocity and acceleration and the scalar for speed: v(t) = i + 2tj a(t) = 2j b.

6. Example 2: The position vector describes the path of an object moving in the xy-plane. a.Sketch a graph of the path. b.Find the velocity, speed, and acceleration of the object at any time, t. c.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.

7. c. The point (3,0) corresponds to t = 0. You can find this by solving: 3cos t = 3 cos t = 1 t = 0 At t = 0, the velocity vector is given by v(0) = 2j, and the acceleration vector is given by a(0) = -3i 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 + 2costj a(t) = -3costi-2sintj r(t) = 3costi + 2sintj

8. Example 3: The position vector r describes the path of an object moving in space. Find the velocity, acceleration and speed of the object. 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:

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