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: $v<em>x = \frac{\Delta x}{\Delta t}$ where $\Delta x = x</em>f - x_i$ (displacement) and $\Delta t$ is the time interval.
• An example: For horizontal motion, $v<em>x = \frac{\Delta x}{\Delta t}$; for vertical motion, $v</em>y = \frac{\Delta y}{\Delta t}$.

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.