Describing Motion in 3-D (and 2-D) §3.4–3.5

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

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

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

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

Position and Displacement Position is where something is r = (x, y, z) Displacement is change in position Dr = (Dx, Dy, Dz) = Dxx + Dyy + Dzz Distance is the magnitude of displacement.

Velocity and Acceleration Velocity v = rate of change of r average vavg = Dr/Dt t is scalar; vavg and Dr are same direction instantaneous v = lim Dr/Dt Dt   0 Speed is the magnitude of velocity. Acceleration a = rate of change of v Average aavg = Dv/Dt instantaneous a = lim Dv/Dt Dt   0

Describing 3-Vectors Position r = x x + yy + zz Velocity v = Dr /Dt = Dx/Dt x + Dy/Dt y + Dz/Dt z = vx x + vy y + vzz Acceleration a = Dv/dt = ax x + ayy + azz

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

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.

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

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. The component of a perpendicular to v causes the direction to change.

Question The rate of change of an object’s speed is the same as the magnitude of its acceleration. Always. Sometimes. Never.