Lecture 1: Motion in One Dimension – Foundations of Kinematics

Course Introduction and Semester Preview

Overview of Physics

Physics is the most fundamental of the sciences, focusing on the behavior and structure of matter. It seeks to explain how the universe operates, from the smallest particles to the largest galaxies.

• Required for major: Essential for many STEM fields.

• Understanding the world: Physics explains natural phenomena and technological advances.

• Enjoyment: Many find the study of physics intellectually rewarding and fun.

Topic Overview

Main Areas in Physics

• Mechanics: Motion of macroscopic objects, forces, friction, circular motion, energy, momentum, planetary motion, rotational motion, oscillations, and waves.

• Fluids: Study of liquids and gases.

• Thermodynamics: Heat, temperature, and energy transfer.

Essential Tools and Math Skills

Units and Conversions

• SI system of units: Standard units for scientific measurement.

• Scientific notation: Expressing very large or small numbers.

• Prefixes: micro-, milli-, centi-, kilo-, etc.

• Estimates: Approximating values for calculations.

Mathematical Foundations

• Linear and quadratic equations

• Basic trigonometry: SOHCAHTOA, Pythagoras

• Calculus: derivatives and integrals

Kinematics: Describing Motion

Key Quantities in Motion

• Position: Where is the object?

• Velocity: How fast and in which direction is it moving?

• Acceleration: How do speed and direction change?

Position

Defining Position

• Position is referenced to a coordinate system.

• Numerical value: x

• Function of time: x(t)

• Initial position: $x_0 = x(t_0)$ (not necessarily zero)

Showing Position

• Use an x-axis with an origin and positive direction.

• Mark positions at specific times.

Position versus Time Graphs

Graphical Representation

• Plot position (x) on the vertical axis and time (t) on the horizontal axis.

• Shows how position changes over time.

Displacement

Definition and Properties

• Displacement: Change in position.

Formula:

$\Delta x = x_f - x_i$

• $\Delta x$ can be positive or negative, indicating direction.

• Displacement is not the same as distance traveled.

Speed and Velocity

Instantaneous and Average Values

• Instantaneous speed: Rate at which position changes, regardless of direction.

• Instantaneous velocity: Speed with direction (vector quantity).

• Average speed: Total distance divided by total time.

• Average velocity: Displacement divided by time interval.

Formula for average velocity:

$\overline{v}_x = \frac{\Delta x}{\Delta t}$

• Unit: $\frac{m}{s}$

• $\overline{v}_x > 0$: object moves in positive x-direction

• $\overline{v}_x < 0$: object moves in negative x-direction

Average Velocity and x-t Graphs

Graphical Interpretation

• Average velocity is the slope between two points on a position-time graph.

Formula:

$\overline{v}_x = \frac{x_3 - x_1}{t_3 - t_1}$

• Negative slope indicates movement to smaller x values.

Instantaneous Velocity

Definition and Calculation

• Instantaneous velocity is the slope of the tangent to the position-time curve at a specific time.

Formula:

$v_x = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$

• Speedometer shows the absolute value of instantaneous velocity.

• $v = |v_x|$ (speed, always positive)

Direction of Velocity

Interpreting Signs

• $v_x > 0$: object moves in the positive x-direction

• $v_x < 0$: object moves in the negative x-direction

Acceleration

Definition and Properties

• Acceleration is the rate at which velocity changes with time.

• It is the slope of the velocity-time graph.

Formula:

$a_x = \frac{dv_x}{dt} = \frac{d^2x}{dt^2}$

• Unit: $\frac{m}{s^2}$

• Acceleration produces change in velocity: $dv_x = a_x dt$

Signs of Acceleration and Velocity

Interpreting Physical Meaning

• If $a_x > 0$ and $dv_x = a_x dt > 0$:

  • If $v_x > 0$: object speeds up

  • If $v_x < 0$: object slows down

• If $a_x < 0$ and $dv_x = a_x dt < 0$:

  • If $v_x > 0$: object slows down

  • If $v_x < 0$: object speeds up

Motion Diagrams

Visualizing Motion

• Motion diagrams help visualize position, velocity, and acceleration over time.

• Example parameters: $x_0 = 0$, $v_{0x} = +4 \frac{m}{s}$, $a_x = -2 \frac{m}{s^2}$

Summary Table: Key Kinematic Quantities

Quantity	Definition	Formula	Unit
Position	Location in space	$x$	m
Displacement	Change in position	$\Delta x = x_f - x_i$	m
Velocity	Rate of change of position	$v_x = \frac{dx}{dt}$	$\frac{m}{s}$
Acceleration	Rate of change of velocity	$a_x = \frac{dv_x}{dt}$	$\frac{m}{s^2}$

Additional info: The notes also reference the importance of scientific notation, SI units, and basic math skills, which are foundational for all physics topics. Motion diagrams and online simulations (e.g., PhET) are recommended for visualizing concepts.