Section 12.3 – Velocity and Acceleration

1. Section 12.3 – Velocity and Acceleration

2. Vector Function A vector function is a function that takes one or more variables and returns a vector: Where and are called the component functions. A vector function is essentially a different notation for a parametric function.

3. Particle Motion In AP Calculus AB, particle motion was defined in functions of time versus motion on a horizontal or vertical line. In AP Calculus BC, particle motion will ALSO be defined in functions of position versus position (along a curve). How successful you are with Particle Motion is a good predictor of how successful you will be on the AP Test.

4. Position Vector Function When a particle moves on the xy-plane, the coordinates of its position can be given as parametric functions: for The particle’s position can also be expressed as a position vector function: The coordinates of the parametric function at time t… … are equal to the components of the vector function at time t. A vector function is essentially a different notation for a parametric function.

5. Example 1 Let the position vector of a particle moving along a curve is defined by (a) Find and graph the position vector of the particle at . y x

6. Velocity Vector Function The vector function for position is differentiable at if and have derivatives at . The derivative of , , is defined as the velocity vector: A vector function is essentially a different notation for a parametric function.

7. Example 1 Continued Let the position vector of a particle moving along a curve is defined by (b) Find the velocity vector. y x

8. Example 1 Continued Let the position vector of a particle moving along a curve is defined by (c) Find and graph the velocity vector of the particle at . y x If the initial point of the velocity vector is also the terminal point of the position vector, the velocity vector is tangent to the curve.

9. Acceleration Vector Function The second derivative of , , is defined as the acceleration vector: A vector function is essentially a different notation for a parametric function.

10. Example 1 Continued Let the position vector of a particle moving along a curve is defined by (b) Find the acceleration vector. y x

11. Arc Length and Speed Consider a particle moving along a parametric curve. The distance traveled by the particle over the time interval is given by the arc length integral: On the other hand, speed is defined as the rate of change of distance traveled with respect to time, so by the Second Fundamental Theorem of Calculus: This is the magnitude of the velocity vector!

12. Speed with a Vector Function The particle’s speed is the magnitude of , denoted : Speed is a scalar, not a vector.

13. Reminder From AP Calculus AB: Speed is the absolute value of velocity: Integrating speed gives total distance traveled: Example: If , find the speed at and the total distance traveled during . THE SAME AS VECTOR FUNCTIONS!

14. White Board Challenge A particle moves along a curve so that and . What is the speed of the particle when .

15. Example 2 A particle moves along the graph of , with its x-coordinate changing at the rate of for . If , find (a) The particle’s position at . You can use the FTOC on components:

16. Example 2 Continued A particle moves along the graph of , with its x-coordinate changing at the rate of for . If , find (b) The speed of the particle at . Since y is a function of x, we need to use the Chain Rule: Find the speed equation: We know dx/dt. Substitute this and dx/dt into the speed equation. From (a), we know x=1.75 when t=2 Substitute t=2

17. Example 3 A particle moves a long a curve with its position vector given by for . Find the time when the particle is at rest. The particle is at rest when the velocity vector is: Find the velocity vector: Solve: is when the particle is at rest because it is the only time on the interval when BOTH components are 0.

18. Summary If is the position vector of a particle moving along a smooth curve in the xy-plane, then, at any time t, • The particle’s velocity vector is ; if drawn from the position point, it is tangent to the curve. • The particle’s speed along the curve is the length of the velocity vector, . • The particle’s acceleration vector is , is the derivative of the velocity vector and the second derivative of the position vector.