Kinematics in Two Dimensions

Chapter 4: Kinematics in Two Dimensions

Chapter Goal

• Learn to solve problems about motion in a plane.
• This chapter integrates kinematics from Chapter 2 and mathematical tools from Chapter 3 to analyze two- and three-dimensional motions.

Main Topics Covered

• Acceleration
• Kinematics in Two Dimensions
• Projectile Motion
• Relative Motion
• Uniform Circular Motion
• Velocity and Acceleration in Uniform Circular Motion
• Nonuniform Circular Motion and Angular Acceleration

Key Concepts

Reading Quizzes

• Projectile Velocity Components: If the velocity vector of a projectile makes an angle θ with a horizontal axis, then the x-component of the velocity is:

  • Answer: A. $v imes ext{sup}(θ)$

• Projectile Acceleration: The acceleration of a particle in projectile motion is:

  • Answer: – directed down at all times.

Case Study: The Hunter and the Coconut

• A hunter aims his rifle directly at a coconut falling from a tree. The bullet and the coconut fall simultaneously.
  • Options:
  • A. The bullet passes above the coconut.
  • B. The bullet hits the coconut.
  • C. The bullet passes beneath the coconut.
  • D. This wasn’t discussed in Chapter 4.

Projectile Motion Overview

• Definition: A projectile is an object that moves in two dimensions under the influence of gravity alone.
• Defining Properties:
  1. Launched with initial velocity $v_0$.
  2. Motion thereafter influenced only by gravity, resulting in free fall.

Mathematical Analysis

Displacement and Velocity

• Position Vector: The position vector of a particle is defined as:
  $extbf{r} = (x, y) = x extbf{i} + y extbf{j}$
• Displacement: The displacement of particle motion can be expressed as: $riangle r = r<em>2 - r</em>1 = (x<em>2 - x</em>1) extbf{i} + (y<em>2 - y</em>1) extbf{j}$
  • Example Calculation: If $r1 = (1 extbf{i} + 1 extbf{j})$ and $r2 = (2 extbf{i} + 2 extbf{j})$, then
    $riangle r = (2 extbf{i} + 2 extbf{j}) - (1 extbf{i} + 1 extbf{j}) = (1 extbf{i} + 1 extbf{j})$
• Average Velocity: Can be computed using: v_{av} = rac{ riangle r}{ riangle t}
  • Instantaneous velocity is defined as the tangent vector to the curve of motion:
    v = rac{d extbf{r}}{dt}

Acceleration

• Acceleration Vector: Defined as the change in velocity over time:
  a = rac{ riangle v}{ riangle t}
• Components of Acceleration must be analyzed:
  • $a = a<em>{||} + a</em>{⊥}$ where:
  • $a_{||}$ is parallel to velocity (affects speed).
  • $a_{⊥}$ is perpendicular to velocity (affects direction).

Kinematics Equations in Two Dimensions

• If acceleration is constant,
  1. $v<em>f = v</em>i + a riangle t$
  2. rf = ri + v_i riangle t + rac{1}{2} a ( riangle t)^2
  3. $v<em>f^2 = v</em>i^2 + 2a riangle r$

Two-Dimensional Kinematics: Projectile Motion

• Two Independent Motions: Horizontal and vertical components can be analyzed separately:
  • Horizontal motion: constant velocity and no acceleration.
  • Vertical motion: undergoes free-fall with acceleration equal to $g$ (gravity).
• Key equations are split across dimensions.
  • Horizontal (x-direction):
  • $x<em>f = x</em>i + v<em>{ix} riangle t$ (where $ax = 0$)
  • Vertical (y-direction):
  • yf = yi + v_{iy} riangle t - rac{1}{2} g ( riangle t)^2

Range and maximum height

• The ideal projectile launch angle for maximum range is θ = 45°.
• Range calculation can involve solving equations for horizontal motion and vertical motion together:
  • Example Range: R = rac{v_i^2 imes ext{sin}(2 heta)}{g}
  • Maximum height occurs when vertical velocity component equals zero.

Relative Motion

• Defined concerning reference frames.
• Key Relationships:
  • If object A moves with velocity $v{A}$ relative to reference frame B, the velocity of object A in frame C would be: $v</em>{AC} = v<em>{AB} + v</em>{BC}$
  • Galilean transformation applies here:
  • If reference frames move relative to each other, velocities can be added or subtracted accordingly.

Circular Motion

• Uniform Circular Motion: An object moving in a circle at constant speed.
  • Tangential speed $v = r imes heta$ where $r$ is the radius and $ heta$ is the angular velocity.
  • The centripetal acceleration ($ac$) is calculated as: ac = rac{v^2}{r}
• Non-Uniform Circular Motion: Angular acceleration is present, leading to changing speeds on the circular path.

Summary of Key Principles

1. Instantaneous Velocity extbf{v} = rac{d extbf{r}}{dt}
2. Instantaneous Acceleration extbf{a} = rac{d extbf{v}}{dt}
3. Kinematic Equations for Constant Acceleration in multiple dimensions follow form similar to traditional physics equations for linear motion.

Important Equations and Relationships

• In projectile motion context, horizontal and vertical motions are analyzed as follows:
• Horizontal:
  • $x<em>f = x</em>i + v_{ix} (t)$
• Vertical:
  • yf = yi + v_{iy} (t) - rac{1}{2} g t^2
• The approximate time of flight is independent of the horizontal component of motion.

Additional Notes on Circular Motion

• Centripetal Acceleration: Always points towards the center of the circular path.
• Angular Velocity: Rate of change of angular position,
  • ext{Average } ω = rac{Δθ}{Δt}
  • Constant angular velocity implies uniform circular motion without changing speed.
• Non-uniform circular motion leads to updates in angular speed often modeled with equations akin to those for linear motion with constant acceleration.
• Key to understanding circular motion dynamics lies in its essential components: tangential and centripetal accelerations