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

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

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

Vectors Position is a vector Velocity is a vector Acceleration is a vector

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

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

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

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

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

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

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

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

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

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

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

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

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)

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.

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