Motion in a Straight Line

• Introduction

  • Rectilinear Motion (Motion along a straight line)
  • Frame of Reference (Coordinate system to locate position)
  • Point Object Assumption (Size negligible compared to distance)

• Position, Path Length, and Displacement

  • Position
    • Specified using time ($t$) and coordinate ($x$)
  • Path Length (Distance)
    • Total length of the path traversed
    • Scalar quantity
    • Always positive
  • Displacement
    • Change in position ($\Delta x = x_2 - x_1$)
    • Vector quantity
    • Can be positive, negative, or zero
    • Magnitude $\le$ Path Length

• Average Speed and Average Velocity

  • Average Speed
    • Total path length / Total time interval
    • Scalar quantity
  • Average Velocity
    • Total displacement / Total time interval ($\Delta x / \Delta t$)
    • Vector quantity

• Instantaneous Speed and Instantaneous Velocity

  • Instantaneous Velocity
    • Velocity at a specific instant
    • Mathematical limit: $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\dt} = \frac{dx}{dt}$
    • Slope of position-time graph
  • Instantaneous Speed
    • Magnitude of instantaneous velocity

• Acceleration

  • Average Acceleration
    • Change in velocity / Time interval ($a_{avg} = \Delta v / \Delta t$)
  • Instantaneous Acceleration
    • Acceleration at a specific instant ($a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$)
    • Slope of velocity-time graph
  • Types of Acceleration
    • Positive Acceleration (Speeding up in positive direction)
    • Negative Acceleration / Retardation (Slow down)
    • Zero Acceleration (Uniform velocity)

• Kinematic Equations for Uniformly Accelerated Motion

  • Condition: Constant acceleration ($a = \text{constant}$)
  • First Equation (Velocity-Time)
    • $v = u + at$
  • Second Equation (Position-Time)
    • $s = ut + \frac{1}{2}at^2$
  • Third Equation (Velocity-Displacement)
    • $v^2 = u^2 + 2as$
  • Displacement in $n^{th}$ Second
    • $s_n = u + \frac{a}{2}(2n - 1)$

• Graphical Representation of Motion

  • Position-Time ($x-t$) Graphs
    • Slope = Velocity
    • Straight line = Uniform motion
    • Parabolic curve = Accelerated motion
  • Velocity-Time ($v-t$) Graphs
    • Slope = Acceleration
    • Area under curve = Displacement
  • Acceleration-Time ($a-t$) Graphs
    • Area under curve = Change in velocity

• Relative Velocity

  • Definition
    • Velocity of an object with respect to another
  • Formula (1D)
    • Velocity of object A relative to B: $v_{AB} = v_A - v_B$
  • Special Cases
    • Same direction: $v_{AB} = v_A - v_B$
    • Opposite direction: $v_{AB} = v_A + v_B$