2.1- Describing Motion

Introduction to Kinematics

  • Focus on describing motion using various representations: motion diagrams, graphs, and mathematical equations.
  • Kinematics: the branch of physics that describes motion, derived from the Greek word kinema, meaning "movement."
  • Related English term: cinema (motion pictures).

Representing Position

  • Kinematic variables (e.g., position, velocity) are measured concerning a coordinate system.
  • Adopt a coordinate system:
    • x-axis for horizontal motion.
    • y-axis for vertical motion.
    • Convention: positive x-direction is to the right, and positive y-direction is upwards.
    • Illustrated in FIGURE 2.1.
Practical Problem: Motion Diagram
  • Example: A student walking to school demonstrates horizontal motion, described using variable x.
  • Coordinate System:
    • Origin (x = 0) is set at the student's starting position, measured in meters.
    • Velocity vectors connect successive positions in the motion diagram, as discussed in Chapter 1.
Analyzing the Motion of the Student
  • Motion diagram indicates the following:
    • Departure at t = 0 min (starting point).
    • Steady progress is made initially; then distance characters vary as follows:
    • At t = 3 min, the distance per interval shortens (possibly slowing down to talk).
    • At t = 6 min, distances in intervals grow longer (possibly speeding up as the student realizes they're late).
Position Data

  • Every dot in the motion diagram represents the student's position at a specific time.

  • TABLE 2.1: Measured Positions of the Student Walking to School:

    • 0\text{ min}: 0\text{ m}
    • 1\text{ min}: 60\text{ m}
    • 2\text{ min}: 120\text{ m}
    • 3\text{ min}: 180\text{ m}
    • 4\text{ min}: 200\text{ m}
    • 5\text{ min}: 220\text{ m}
    • 6\text{ min}: 240\text{ m}
    • 7\text{ min}: 340\text{ m}
    • 8\text{ min}: 440\text{ m}
    • 9\text{ min}: 540\text{ m}

  • The motion can also be represented graphically:

    • FIGURE 2.3: Graph of the student's positions (x vs. t).
    • Definition: A graph of "a versus b" has a on the vertical axis and b on the horizontal axis. Represents "a as a function of b."

  • Continuous curve assumption:

    • Represents motion across all intervening points of space as a continuous position-versus-time graph (refer to FIGURE 2.4).
    • Note: Graph is abstract, not a direct representation of motion.

Analyzing Velocity

  • Velocity is a vector with both magnitude and direction, indicated as v.
  • In one dimension:
    • Velocity vectors are restricted to forward/backward for horizontal; up/down for vertical.
    • Notation for horizontal motion: v_x is positive when moving right, negative when left.
    • Notation for vertical motion: v_y for vertical direction.
Speed vs. Velocity
  • Defined: speed is the magnitude of velocity (always positive).
  • Velocity equation: vx = \frac{\Delta x}{\Delta t} where \Delta x = xf - x_i (displacement) and \Delta t is the time interval.
  • An example: For horizontal motion, vx = \frac{\Delta x}{\Delta t}; for vertical motion, vy = \frac{\Delta y}{\Delta t}.
Position-Versus-Time Graph Interpretation
  • Examining FIGURE 2.8:
    • Motion diagram shows three phases:
    • Constant speed in the first phase.
    • Decreased speed in the second phase.
    • Increased speed in the final phase.
  • Relating graph slopes to speed:
    • Faster speed corresponds to steeper slopes on the position-versus-time graph (refer to FIGURE 2.9).
    • Slope definition: ratio of "rise" to "run" determines the velocity at the given position.

Transitioning Between Position and Velocity

From Velocity to Position
  • If a velocity graph is given, how to determine the position graph?
  • Example: Leaving a lecture and walking towards class:
    • First phase (walking away): velocity +1.0\text{ m/s}.
    • Second phase (running back): velocity -3.0\text{ m/s}.
  • Graph Analysis:
    • Figures depict how a constant velocity leads to a corresponding slope on the position graph, revealing the path taken.
    • As conditions change, analyze the segment by segment, depicting how position is attained based on the velocity graphs.
    • Initial position: Often set where you start at x = 0\text{ m}.

Concluding Summary

  • Understanding the relationship between velocity and position reveals kinetic principles in motion representation, facilitating problem-solving in kinematics.
  • Techniques to interpret graphical data reinforce precision in analyzing motion through visual representation and mathematical equations.