Kinematics in Two Dimensions Study Notes

Kinematics in Two Dimensions (Chapter 4)

Acceleration Vectors (4.1)

• The acceleration vector, $\vec{a}_{avg}$, always points in the same direction as the change in velocity vector, $\Delta \vec{v}$:
  • $\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}$
• The velocity vector, $\vec{v}$, can change in two ways:
  • By changing its magnitude (speed).
  • By changing its direction.
• Parallel Component of Acceleration ($\vec{a}_{||}$): When the magnitude of the velocity vector (speed) changes, it is caused by the parallel component of acceleration.
  • If only the speed changes, then the perpendicular component of acceleration is zero: $\vec{a}_{\perp} = 0$.
• Perpendicular Component of Acceleration ($\vec{a}_{\perp}$): When the direction of the velocity vector changes, it is caused by the perpendicular component of acceleration.
  • If only the direction changes, then the parallel component of acceleration is zero: $\vec{a}_{||} = 0$.
• Examples of Acceleration Direction:
  • If a particle has steady speed and is curving upward, its acceleration is perpendicular to its velocity, pointing generally in the direction of the curve (e.g., North if moving East and curving up).
  • If a particle is slowing down and curving to the left, its acceleration will have both a component opposite to its velocity (for slowing down) and a component perpendicular to its velocity, inward to the curve (for changing direction). The resultant vector will encompass both effects.

Acceleration in Circular Motion (4.1)

• An object in uniform circular motion has a constant speed.
• However, the continuous change in the direction of its velocity produces a