Chapter: Motion in a Plane

• Introduction

  • Definition of motion in two dimensions ($x$ and $y$ axes)
  • Examples: Projectile motion, circular motion
  • Distinction between scalars and vectors

• Scalars and Vectors

  • Scalar Quantities: Magnitude only (e.g., mass, speed, time)
  • Vector Quantities: Magnitude and direction (e.g., displacement, velocity, force)
  • Representation: Directed line segments ($\vec{A}$)
  • Equal and negative vectors

• Vector Addition and Subtraction

  • Graphical Method
    • Head-to-tail method
    • Triangle Law of Vector Addition
    • Parallelogram Law of Vector Addition ($R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$)
  • Analytical Method
    • Resolution of vectors into rectangular components ($A_x = A\cos\theta$, $A_y = A\sin\theta$)
    • Unit vectors ($\hat{i}, \hat{j}, \hat{k}$)
  • Multiplication of Vectors
    • Dot Product / Scalar Product ($\vec{A} \cdot \vec{B} = AB\cos\theta$)
    • Cross Product / Vector Product ($\vec{A} \times \vec{B} = AB\sin\theta \hat{n}$)

• Motion in a Plane (Kinematics)

  • Position Vector: $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}$
  • Displacement: $\Delta\vec{r} = \Delta x\hat{i} + \Delta y\hat{j}$
  • Average Velocity: $\vec{v}_{avg} = \frac{\Delta\vec{r}}{\Delta t}$
  • Instantaneous Velocity: $\vec{v} = \frac{d\vec{r}}{dt}$ (Tangent to the path)
  • Acceleration: $\vec{a} = \frac{d\vec{v}}{dt}$

• Projectile Motion

  • Definition: Motion under the influence of gravity alone
  • Assumptions: Air resistance neglected
  • Horizontal Motion: Constant velocity ($a_x = 0$)
  • Vertical Motion: Constant acceleration ($a_y = -g$)
  • Key Formulas
    • Trajectory Equation: $y = (\tan\theta_0)x - \frac{g}{2(v_0\cos\theta_0)^2}x^2$ (Parabolic path)
    • Time of Flight ($T$): $\frac{2v_0\sin\theta_0}{g}$
    • Maximum Height ($H$): $\frac{v_0^2\sin^2\theta_0}{2g}$
    • Horizontal Range ($R$): $\frac{v_0^2\sin(2\theta_0)}{g}$

• Uniform Circular Motion

  • Definition: Motion along a circular path with constant speed
  • Angular Displacement ($\Delta\theta$) and Angular Velocity ($\omega = \frac{d\theta}{dt}$)
  • Relation between linear and angular velocity ($v = r\omega$)
  • Centripetal Acceleration ($a_c = \frac{v^2}{r} = r\omega^2$)
    • Always directed towards the center of the circular path
  • Centripetal Force: Required force to maintain circular motion ($F_c = \frac{mv^2}{r}$)