Motion in One Dimension - Part 2

Jul 20, 2024

Lecture: Motion in One Dimension (Part 2)

Overview of Previous Video

  • Covered: distance, displacement, speed, velocity (instantaneous and average), and important graphs.

Topics Covered

  1. Acceleration
  2. Equations of Motion
  3. Motion under Gravity
  4. Relative Velocity
  5. Two-Dimensional Problems

1. Acceleration

Definition

  • Acceleration: Rate of change of velocity with time
  • Uniform Acceleration: Velocity changes by equal amounts in equal intervals of time
  • SI Unit: m/s²
  • Dimensions: L¹ T⁻²

Types of Acceleration

  • Positive Acceleration: Velocity increases with time
  • Negative Acceleration (Retardation/Deceleration): Velocity decreases with time
  • Zero Acceleration: Velocity is constant, state of uniform motion
  • Key Concept: Zero velocity does not imply zero acceleration

Examples

  • Throwing a Particle Upwards: Zero velocity at highest point, but acceleration due to gravity (g)
  • Acceleration and Velocity Relation:
    • In the same direction: Speed increases
    • Opposite direction: Speed decreases

2. Average and Instantaneous Acceleration

Definitions

  • Average Acceleration: Change in velocity over a time interval (ΔV/ΔT)
  • Instantaneous Acceleration: Limit of average acceleration as ΔT approaches 0, mathematically (dV/dT)

Graphical Representation

  • Average Acceleration: Slope of a straight line on a velocity-time graph (ΔV/ΔT)
  • Instantaneous Acceleration: Slope of the tangent to the velocity-time graph at a point

3. Equations of Motion

Derivations

  1. Velocity-Time Relation: V = U + AT
  2. Displacement-Time Relation: S = UT + ½AT²
  3. Velocity-Displacement Relation: V² = U² + 2AS
  4. Distance Travelled in the Nth Second: SN = U + A/2(2N - 1)

Motion under Gravity

  • Constant Acceleration: G
  • Sign Conventions: Depends on direction chosen for positive and negative

Free Falling Body

  • U = 0, S = H, A = G
  • Calculations for time of fall, velocity, etc.

Projected Upward

  • Initial velocity (U), max height (H), A = -G
  • Time and height calculations

4. Relative Velocity

  • Definition: Velocity of one object relative to another
  • Formulas:
    • V(AB) = V(A) - V(B)
    • V(BA) = V(B) - V(A)
    • Relation between magnitudes

Examples

  • Bus and Cyclist Example: Calculation of relative speed and the period of the bus service

Position-Time Graphs

  • Analysis of relative velocity using position-time graphs
  • Different scenarios based on relative motion and graph slopes

5. Two-Dimensional Problems

Examples and Solutions

  • Walking Direction Problem: Calculation of distance and displacement using vector addition
  • Velocity-Time Graph Problems: Determining displacement and distance from areas under graph curves
  • Problems involving different speeds: Average speed, relative speed calculations

Final Notes

  • Emphasis on understanding underlying principles

  • Practice with numerical problems to solidify concepts

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