Chapter 4A

Chapter Overview

• Chapter 4: Kinematics in Two Dimensions

• Focus: Solving problems about motion in a plane.

Motion in a Plane

Trajectory

• A particle’s motion is represented in the xy-plane through its trajectory.

• The position vector helps to define the particle's location.

• Graphs will show y versus x, depicting actual movement rather than abstract representation.

Average and Instantaneous Velocity

• Average Velocity: Direction of displacement over a time interval.

• Instantaneous Velocity: Limit of average velocity as the time interval approaches zero.

  • Tangent to the trajectory at the given time.

  • Mathematically defined as ( v = \lim_{\Delta t \to 0} \frac{\Delta r}{\Delta t} )

Velocity Components

• Velocity can be expressed in terms of its components:

  • If the angle ( \theta ) is measured from the positive x-direction:

    • ( v_x = v \cos(\theta) )

    • ( v_y = v \sin(\theta) )

Acceleration

Definition

• Defined as the change in velocity over time, represented as a vector.

• Important points of change: magnitude (speed) or direction.

Finding the Acceleration Vector

• To find the average acceleration between two velocity points, draw the velocity vectors and apply vector subtraction.

Components of Acceleration

• Acceleration can be decomposed into:

  • Parallel Component: Affects speed.

  • Perpendicular Component: Affects direction.

• Important for analyzing the motion and determining changes in trajectory.

Instantaneous Acceleration

• The instantaneous acceleration is calculated as the limit of average acceleration where both velocity vectors approach each other:

  • ( a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} )

  • Found at the same point on the trajectory.

Decomposing Acceleration

• The acceleration vector has components ( a_x ) and ( a_y ):

  • Relating to changes in speed and direction respectively.

• If acceleration is constant, both components stay constant and can be analyzed independently for motion analysis.

Projectile Motion

Definition

• A projectile moves in two dimensions under gravity, ignoring air resistance, following a parabolic trajectory.

  • Launched with initial velocity at a specific angle above the x-axis.