Motion in a Plane - Class 11 Physics

Motion in a Plane

1. Introduction

Motion in a plane involves motion in two dimensions (2D), like motion on a flat surface.

2. Scalars and Vectors

  • Scalar: Quantity with magnitude only (e.g., speed, mass, time).
  • Vector: Quantity with both magnitude and direction (e.g., velocity, force).

3. Vector Representation

  • Represented by an arrow.
  • Length of arrow = magnitude, direction = direction of vector.
  • Notation: →A or A̅.

4. Vector Addition

  • Triangle Law: Place tail of second vector at head of first.
  • Parallelogram Law: Both vectors from a common point. Diagonal is the sum.

5. Components of a Vector

Any vector can be broken into two perpendicular components:

  • Horizontal (x-axis): Ax = A cosθ
  • Vertical (y-axis): Ay = A sinθ
  • Resultant vector: A = √(Ax² + Ay²)

6. Motion in Two Dimensions

Position, velocity, and acceleration are vector quantities and can be resolved into components along x and y axes.

7. Projectile Motion

  • A type of 2D motion under constant acceleration due to gravity.
  • Path: Parabolic

Important Equations:

  • Time of Flight: T = (2u sinθ)/g
  • Maximum Height: H = (u² sin²θ)/(2g)
  • Horizontal Range: R = (u² sin2θ)/g

Where:

  • u: Initial speed
  • θ: Angle of projection
  • g: Acceleration due to gravity

8. Uniform Circular Motion

  • Object moves in a circle with constant speed.
  • Direction of velocity changes → acceleration present.
  • Acceleration is called centripetal acceleration.

Formulas:

  • Speed: v = 2πr / T
  • Centripetal Acceleration: a = v² / r
  • Centripetal Force: F = mv² / r

9. Relative Velocity in 2D

Used when both objects move in 2D. Vector subtraction is used to find relative velocity.

Formula: VAB = VA - VB

10. Vector Operations

  • Addition: Triangle or parallelogram method
  • Subtraction: Add the negative vector
  • Multiplication with scalar: Changes magnitude, not direction
  • Dot Product: A · B = AB cosθ (Scalar)
  • Cross Product: A × B = AB sinθ (Vector perpendicular to plane)

11. Key Concepts Summary

  • Vectors have direction; scalars do not.
  • Projectile motion is a combination of horizontal uniform motion and vertical uniformly accelerated motion.
  • Uniform circular motion involves continuous acceleration toward the center (centripetal).
  • Vectors follow geometrical rules for addition and subtraction.