1. Describing Motion
in 3-D
(and 2-D)
§3.1–3.2

2. Poll Question
Two objects can travel at the same speed but have different velocities.
True
False

3. Poll Question
Two objects can travel at the same velocity but have different speeds.
True
False

4. Vectors Position is a vector Velocity is a vector
Acceleration is a vector

5. Describing 3-Vectors Position r = x i + y j + z k Velocity v = dr/dt
= dx/dt i + dy/dt j + dz/dt k
= vx i + vy j + vz k
Acceleration a = dv/dt
= d2x/dt2 i + d2y/dt2 j + d2z/dt2 k
= ax i + ay j + az k

6. Magnitudes of 3-Vectors
Distance from origin r = | r | = x2 + y2 + z2
Speed v = | v | = vx2 + vy2 + vz2
Magnitude of acceleration a = | a | = ax2 + ay2 + az2

7. Change Vectors Position and velocity may be in different directions
Velocity and acceleration may be in different directions
q0 + Dq = qf Dq = qf − q0

8. Change Vectors Position and velocity may be in different directions
Velocity and acceleration may be in different directions
q0 + Dq = qf Dq = qf − q0

9. Quick Question vi vf
What is the direction of Dv? A B C D E F G H

10. Quick Question vi vf
What is the direction of Dv? A B C D E F G H

11. Quick Question vi vf
What is the direction of Dv? A B C D E F G H

12. Quick Question vi vf
What is the direction of Dv? A B C D E F G H

13. Quick Question vi vf
What is the direction of Dv? A B C D E F G H

14. Familiar Situations Ballistic trajectories Circular motion
If a || v, path is straight.
If a || v, path is curved.

15. Poll Question
If an object’s distance from the origin r does not change, its velocity must be zero.
True.
False.

16. Poll Question
If an object’s speed v does not change, its acceleration must be zero.
True.
False.

17. Acceleration and Velocity
The component of a parallel to v causes the speed to change.
a|| = dv/dt
The component of a perpendicular to v causes the direction to change.
formula TBA (when we get to circular motion)

18. Poll Question
The rate of change of an object’s speed d|v|/dt is the same as the magnitude of its acceleration |dv/dt|.
Always.
Sometimes.
Never.

19. r = 15 m/s t i + (15 m/s t – 5 m/s2 t2) j.
Board Work
A baseball thrown from the origin follows a path defined by
r = 15 m/s t i + (15 m/s t – 5 m/s2 t2) j.
Sketch its horizontal position, velocity, and acceleration vs. time.
x-t, vx-t, ax-t
Sketch its vertical position, velocity and acceleration vs. time.
y-t, vy-t, ay-t