One-Dimensional Kinematics, Graphical Analysis, and Dynamics

Higher-Order Motion and Concepts of One-Dimensional Study

• Patterns in Motion Units
  • There is a discernible pattern to the units of motion derivatives:
    • Velocity: Measured in meters per second ($m/s$).
    • Acceleration: The change in velocity over time, measured in meters per second per second, or meters per second squared ($m/s^2$).
    • Jerk: The change in acceleration over time, measured in meters per second per second per second ($m/s^3$).
  • Although these patterns continue ad infinitum, standard physics primarily focuses on position, velocity, and acceleration.
  • Practical Applications of Higher Derivatives: Engineers, rather than pure physicists, often utilize jerk (the third derivative). For example, a roller coaster designer would worry about jerk to ensure a ride is as smooth as possible. However, the fundamental laws of physics are generally expressed in terms of position, velocity, and acceleration.

Movement in One Dimension (1D)

• Simplification Strategy

  • Reality exists in three spatial dimensions ($x, y, z$). However, if an object moves in a straight line, the other dimensions are irrelevant because no motion occurs within them.
  • Aligning the $x$-axis with the straight-line path allows for a simplified description of real-world physical situations.
  • This approach is common in introductory, graduate, and even quantum physics to understand core concepts before generalizing to two or three dimensions.

• Physical Coordinates and Vector Nature

  • When moving only on the $x$-axis, the $y$ and $z$ coordinates are always zero ($y = 0; z = 0$).
  • Consequently, the position vector only has an $x$-component ($r_x$).
  • In one dimension, directionality is limited to positive or negative. As long as signs are handled properly (positive for the forward direction, negative for the reverse), the formal vector notation can often be ignored for simplicity.

• One-Dimensional Kinematic Definitions

  • Instantaneous Velocity ($v_x$): Defined as the change in the $x$ coordinate over the change in time for very small intervals of time ($\Delta t$).
    • Formula: $v_x = \frac{\Delta x}{\Delta t}$
    • The subscript $x$ serves as a reminder that this is the $x$-component of a more general velocity vector.
  • Instantaneous Acceleration ($a_x$): Defined as the change in the $x$-velocity over the change in time for small intervals.
    • Formula: $a_x = \frac{\Delta v_x}{\Delta t}$
  • Calculus Connection: For students who have taken calculus, it is notable that velocity is the derivative of position, and acceleration is the derivative of velocity as $\Delta t$ approaches zero.

Graphical Analysis of Motion

• Mathematical Review of Straight Lines

  • Equation of a line: $y = mx + b$.
    • $b$: The $y$-intercept (the distance between the curve and the origin at $x = 0$).
    • $m$: The slope ($\text{rise} / \text{run}$ or $\Delta y / \Delta x$).
  • Steepness and Sign: A horizontal line has a slope of zero. A 45-degree line has a slope of one. Negative slopes indicate lines tilted backwards vertically.

• Mapping Physics to Math Axes

  • To interpret motion graphically, physical units are mapped to mathematical axes:
    • Position vs. Time Plot: Mathematical $y$ becomes physical position ($x$ in meters). Mathematical $x$ becomes physical time ($t$ in seconds).
    • Velocity vs. Time Plot: Mathematical $y$ becomes physical velocity ($v$ in $m/s$). Mathematical $x$ becomes physical time ($t$ in seconds).

• Slope-Value Relationships

  • Velocity from Position Graphs: The numeric value of an object's velocity at a specific moment equals the slope of the position vs. time plot at that moment.
    • High slope = high velocity (steep curve).
    • Low slope = low velocity.
    • Zero slope = stationary object.
  • Acceleration from Velocity Graphs: The numeric value of an object's acceleration at a specific moment equals the slope of the velocity vs. time plot at that moment.
    • Positive slope = speeding up ($v$ is increasing).
    • Negative slope = slowing down ($v$ is decreasing).
    • Zero slope = constant velocity (no acceleration).
  • Smoothness Assumption: While real motion can be complicated, we treat curves as collections of tiny line segments. Zooming in enough on any point of a smooth curve reveals it to be a straight line.

The Story of a Short Race: Graphical Example

• Stage 1: Motionless at Starting Line

  • Position: Horizontal line at $x = 0$.
  • Velocity: Slope of position is zero, so velocity is zero.
  • Acceleration: Slope of velocity is zero, so acceleration is zero.

• Stage 2: Speeding Up (The Start)

  • Position: Curve bending upwards (increasing slope over time).
  • Velocity: Since slope of position increases, the value of velocity increases (represented as a diagonal line with positive slope).
  • Acceleration: Since the slope of velocity is a positive constant, acceleration is a constant positive value (a flat horizontal line above zero).

• Stage 3: Max Speed (Steady Travel)

  • Position: Straight diagonal line (constant slope as distance increases).
  • Velocity: Since position slope is constant, velocity is a constant positive value (flat horizontal line).
  • Acceleration: Since velocity slope is zero, acceleration returns to zero.

• Stage 4: Slowing Down after Finish

  • Position: Curves and flattens out (slope decreases toward zero).
  • Velocity: Since position slope decreases, velocity graph slopes downward toward zero.
  • Acceleration: Since velocity slope is negative, acceleration is a constant negative value (horizontal line below zero).

• Stage 5: Stopped

  • Position: Flat horizontal line (constant distance).
  • Velocity: Zero.
  • Acceleration: Zero.

Newton's Second Law and Dynamics

• Fundamentals of $F_{net} = ma$

  • A non-zero net force (an overall push or pull) causes acceleration.
  • A strong net force results in high acceleration (rapid change in velocity).
  • A weak net force results in low acceleration (gradual change in velocity).
  • Inertia: The resistance to changes in motion, which is directly related to mass ($m$). Solving for acceleration: $a = \frac{F_{net}}{m}$.

• The Influence of Mass

  • Higher mass objects experience lower acceleration for a given net force. This is why a stalled car in neutral is hard to get moving—not necessarily due to friction, but due to its high mass resisting acceleration.
  • Real-World Application: 18-wheeler trucks have high mass. Because they obey $F_{net} = ma$, they cannot speed up, slow down, or turn as rapidly as small commuter cars. This is a physical limitation, not a behavioral choice by the driver.

• Vehicular Control and Acceleration

  • Acceleration involves speeding up, slowing down, or turning. Most cars have three corresponding controls:
    • Gas Pedal: Forward acceleration vector (speeding up).
    • Brake Pedal: Backward acceleration vector (slowing down).
    • Steering Wheel: Sideways acceleration vector (turning).

Analytical Procedure for Physics Problems

1. Identify Forces: List all forces (gravity, friction, tension, etc.) and create a free-body diagram using arrows.
2. Vector Addition: Sum those forces to find the Net Force ($F_{net}$). Forces in opposite directions subtract; equal and opposite forces cancel out.
3. Calculate Acceleration: Use Newton's second law ($a = \frac{F_{net}}{m}$).
4. Determine Velocity and Position: Use the acceleration to find functions of velocity and position over time. This requires knowing constants like initial velocity ($v_0$) and initial position ($x_0$).

Case Study: Grandma in the Elevator

• Scenario Details

  • Mass ($m$): $100\,kg$.
  • Starting level: Ground ($0^s$).
  • Normal Force ($F_N$) while moving up: $1200\,N$.
  • Note: Scales measure Normal Force, not weight. If stationary, $F_N = \text{weight}$; if moving, they differ.

• Step 1: Calculate Forces

  • Force of Gravity ($F_g$): $100\,kg \times 9.8\,m/s^2 = 980\,N$ (Downward, $-y$ direction).
  • Normal Force ($F_N$): $1200\,N$ (Upward, $+y$ direction).

• Step 2: Calculate Net Force

  • $F_{net} = 1200\,N - 980\,N = 220\,N$ (Upward, $+y$ direction).

• Step 3: Calculate Acceleration

  • $a = \frac{220\,N}{100\,kg} = 2.2\,m/s^2$.
  • Sanity check: $2.2\,m/s^2$ is approximately $0.22\,g$ (G-force). This is reasonable; an elevator shouldn't feel like a high-G roller coaster for a grandmother.

• Step 4: Calculate Velocity

  • If starting at $t = 0$ with $v = 0$:
    • After $1\,s$, $v = 2.2\,m/s$.
    • After $2\,s$, $v = 4.4\,m/s$.
  • Sanity check: $4.4\,m/s$ is roughly $9\,mph$, a safe speed for an elevator.

Constant Acceleration Equations (Kinematics)

• Defining the Constant Case

  • In many problems, acceleration is a simple constant number (like $3\,m/s^2$) rather than a complex function of time.

• Velocity Under Constant Acceleration

  • Derived from the equation of a straight line on a velocity-time graph:
    • $v_x(t) = v_{0x} + a_{0x}t$
  • Where $v_{0x}$ is initial velocity and $a_{0x}$ is the constant acceleration.

• Position Under Constant Acceleration

  • Derived from a quadratic function (parabolic) on a position-time graph:
    • $x(t) = x_0 + v_{0x}t + \frac{1}{2} a_{0x}t^2$
  • This formula accounts for:
    • $x_0$: Initial position (vertical intercept).
    • $v_{0x}$: Initial velocity (the initial slope of the curve).
    • $\frac{1}{2} a_{0x}$: Based on the mathematical concavity of the curve.