Chapter 1: Description of Motion - Notes
Chapter 1: Description of Motion
Overview and Goals
Goals of this Chapter:
Introduce concepts of scalar and vector quantities.
Learn how to quantify, describe, and measure motion in one dimension (1D).
Key Vocabulary:
Scalar
Magnitude
Vector
Frame of reference
Path length
Displacement
Speed
Velocity
Average vs. Instantaneous
Quantities, Variables, and SI Units (Partial Table from source - not fully provided in transcript for all items):
Position
Path length: SI Unit - meter (m)
Displacement: SI Unit - meter (m)
Speed: SI Unit - meters per second (m/s)
Velocity: SI Unit - meters per second (m/s)
Two Types of Quantities
Scalars:
Fully described by only a magnitude.
Magnitude refers to a number, size, or measurement.
Examples: Length, Time, Distance, Speed.
Vectors:
Described by both a magnitude and a direction.
In one dimension (1D), the direction is typically indicated by a positive (+) or negative (−) sign.
Identification:
Handwritten: An arrow is placed above the variable, e.g., \vec{v} .
Printed (or Digital): Bold font is used, e.g., \mathbf{v} .
To describe the magnitude only, the following forms might be used: v, \vec{v} , \mathbf{v} (often referring to the absolute value or magnitude of the vector).
Position
Definition: Position is an object's location in terms of a frame of reference.
Frame of Reference: This defines a starting point or origin for the motion, from which all positions are measured.
Coordinates: Position is typically defined using coordinates, such as x (for horizontal motion) or y (for vertical motion).
Subscripts: Subscripts are used to distinguish between initial (denoted by subscript i) and final (denoted by subscript f) values, e.g., xi, xf .
Path Length (Distance)
Definition: Path length s (or \Delta s) is the total distance of the full path traveled by an object.
Analogy: It is similar to an odometer reading in a car.
SI Unit: meter (m).
Nature: It is a scalar quantity, meaning it only has magnitude and no direction.
Important Symbol \Delta (Delta):
Means "change in."
Expressed as: (\ ){final} - (\ ){initial}.
Displacement
Definition: Displacement \Delta x (sometimes denoted as D in textbooks) is the overall change in position, regardless of the path taken.
SI Unit: meter (m).
Nature: It is a vector quantity, meaning it has both magnitude and direction.
1D Representation:
\Delta x is used for horizontal displacement.
\Delta y is used for vertical displacement.
Formula: \Delta x = xf - xi (where xf is the final position and xi is the initial position).
Example 1: Path Length vs. Displacement
Scenario: An individual leaves a house, goes to a grocery store, then proceeds to a pizzeria, and finally returns home.
Problem 1: Path length (Grocery store to pizzeria to home)
Problem 2: Displacement (Grocery store to pizzeria to home)
Problem 3: Path length (House to grocery store)
Problem 4: Displacement (House to grocery store)
Answer Choices (for a specific scenario with numerical values, though values not provided in transcript for general explanation): A. 3.2 \text{ mi}, B. 5.0 \text{ mi}, C. 7.3 \text{ mi}, D. 0 \text{ mi}, E. none of the above. (These choices imply a specific scenario given in the original presentation).
Speed
Instantaneous Speed: The speed of an object at any specific "split-second" instant of time.
Average Speed \bar{v} or v_{avg}:
Defined as the total path length traveled divided by the total time taken.
It approximates the instantaneous speed over a longer duration.
SI Unit: meters per second (m/s).
Nature: It is a scalar quantity.
Formula for Average Speed: \bar{v} = v{avg} = \frac{s{tot}}{t{tot}} (where s{tot} is total path length and t_{tot} is total time).
Uniform Motion:
Describes motion with constant speed (the speed is the same at any instant).
This is often a simplification, as true motion is typically non-uniform.
Example 2: Speed & Units
Scenario: A car's trip composed of three segments:
Travels at 20.0 \text{ km/h} for 1.35 \text{ h}.
Stops for 1.00 \text{ h}.
Travels at 60.0 \text{ km/h} for 2.00 \text{ h}.
Calculations Required:
a) Distance traveled during the 1st segment (in km).
b) Distance traveled during the 3rd segment (in km).
c) Average speed for the whole trip (in both km/h and SI units).
(Average) Velocity
Definition: Velocity considers both the instantaneous speed and the direction of the displacement.
SI Unit: meters per second (m/s).
Nature: It is a vector quantity.
Formula for Average Velocity \vec{v}_{avg} or \mathbf{\bar{v}} (often shortened to \mathbf{v} or v when discussing average):
\mathbf{\bar{v}} \equiv \frac{\Delta \mathbf{x}}{\Delta t} = \frac{\mathbf{x}f - \mathbf{x}i}{tf - ti} .
If the initial time is ti = 0, the formula simplifies to: \mathbf{\bar{v}} = \frac{\mathbf{x}f - \mathbf{x}i}{tf - 0} = \frac{\mathbf{x}f - \mathbf{x}i}{t} .
Educational Tool Mentioned: The "Moving Man SIM" is a simulation often used to visualize these concepts.
Your Turn: Velocity vs. Speed
Scenario: Two cars, a blue car and an orange car, travel from point P to point Q in equal amounts of time. Their paths might differ (e.g., straight for one, curved for another).
Questions:
Which car has the greatest (average) speed? (Choices: A. Blue car, B. Orange car, C. Both the same).
Which car has the greater (average) velocity? (Choices: A. Blue car, B. Orange car, C. Both the same).
Key Distinction: Speed depends on path length, while velocity depends on displacement.
Example 3: Speed vs. Velocity
Scenario: A person rides a bike:
20 \text{ km} East in 35 \text{ min}.
Then turns around and rides 15 \text{ km} West in 25 \text{ min}.
Calculations Required (in SI units):
a) Average speed.
b) Average velocity.
Note: This requires converting km to m and min to s, and carefully distinguishing between total path length for speed and net displacement for velocity.