Intro to Kinematics: Scalars, Vectors, One-Dimensional Motion
Branches of Physics
Physics has multiple branches: Mechanics, Thermodynamics, Electromagnetism, Optics, etc.
This course focuses on mechanics, with a brief plan to cover thermodynamics as well; electromagnetism and optics are mentioned as other branches not the main focus here.
Within mechanics, there are two subfields: Kinematics and Dynamics.
Kinematics: the study of motion without considering causes (why motion happens).
Dynamics: the study of motion considering the causes (forces and torques) that produce motion.
The course starts with kinematics (how objects move) and postpones dynamics (the forces behind motion) for later.
Mechanics: Kinematics vs Dynamics
Kinematics answers questions like: How much did an object move? What is its displacement, velocity, and acceleration? In which direction?
Dynamics answers questions like: Why did the object move that way? What forces acted on it to produce the motion?
Plan for this course: begin with kinematics, then return to dynamics to explain why motions occur.
Scalars and Vectors in Physics
Quantities in physics can be scalars or vectors:
Scalar quantity: magnitude only (no direction).
Vector quantity: magnitude and direction.
Examples:
Scalars: Time, Temperature, Energy (e.g., time = 2 s, temperature = 86 °F, energy = 50 J).
Vectors: Gravity (a force), Acceleration, Velocity, Displacement.
Key point: Vectors require both a size (magnitude) and a direction; scalars require only magnitude.
Gravity is a vector (a force with direction). Acceleration and velocity are vectors. Displacement is a vector.
Some quantities we encounter can be scalars (time, temperature, energy) and some are vectors (velocity, acceleration, displacement).
Coordinate Systems and Reference Points
Motion is described with respect to a reference point or origin in a coordinate system.
Cartesian coordinate system (2D or 3D) is the simplest commonly used system:
x-axis (horizontal): positive to the right, negative to the left.
y-axis (vertical): positive upward, negative downward (in a chosen orientation).
In many introductory problems, motion is taken as one-dimensional along the x-axis:
The motion is described relative to an origin (reference point) on a line.
The origin O is where the axes intersect.
One-dimensional motion means the object moves along x only (either to the right/positive or to the left/negative).
One-Dimensional Kinematics: Position and Displacement
Position: where the object is relative to the reference origin.
Example: an object initially at some position xi and later at xf.
Displacement (Δx): the change in position, a vector in 1D represented by the signed difference between final and initial positions:
Delta x = xf - xi
In 1D, this is effectively the signed distance along the x-axis.
Positive displacement indicates motion to the right (positive x direction); negative displacement indicates motion to the left (negative x direction).
Magnitude of displacement: |\Delta x| (the distance from initial to final position, irrespective of direction).
Displacement as a vector: in full generality, {Delta x} = {x}f - {x}i; in 1D this reduces to the scalar \Delta x = xf - xi with a sign indicating direction.
Initial position vs final position notation:
Often denote initial position as xi or x\o,andfinalpositionat xf or x.
Distance Traveled vs Displacement
Displacement is the vector from the initial position to the final position (has magnitude and direction).
Distance traveled (path length) is the total length of the path traveled, i.e., the sum of the magnitudes of each segment of the motion:
If motion consists of several segments with displacements (\Delta x1, \Delta x2, \Delta x_3) along the path, then
D = |\Delta x1| + |\Delta x2| + |\Delta x_3| \
This formula indicates that distance is always a positive quantity, reflecting the actual ground covered, regardless of direction. In contrast, displacement refers to the change in position of an object and is defined as the difference between the final position and the initial position, taking direction into account. The formula for displacement can be expressed as:\
\
\Delta x = x{final} - x{initial} \
\
This quantity can be positive, negative, or zero depending on the direction of the motion. In addition, it is crucial to distinguish between displacement and distance traveled, as displacement refers to the change in position while distance is the total path length covered regardless of direction. Understanding this difference is essential as it impacts how we analyze and interpret motion in various scenarios. For example, while two objects may travel the same distance, their displacements can differ greatly based on their starting and ending points. Furthermore, displacement is a vector quantity, meaning it has both magnitude and direction, whereas distance is a scalar quantity that only has magnitude. To further explore these concepts, we can analyze various motion scenarios, such as an object moving in a straight line with constant speed versus one that accelerates, to see how they affect the calculations of both distance and displacement. When examining one-dimensional motion, it is also important to consider the definitions of speed and velocity, where speed is the rate at which an object covers distance and is a scalar quantity, while velocity is defined as the rate of change of displacement and includes direction, making it a vector quantity. By differentiating between these quantities, we can better understand how they influence the behavior of moving objects and apply this knowledge to real-world situations, such as vehicles on a highway or projectiles in motion.
Distinction illustrated:
You can have small net displacement but large distance traveled if you move back and forth.
Examples from the transcript:
Example 1: initial at 2 m, final at 4 m.
Delta x = xf - xi = 4{ m} - 2{ m} = +2{ m}(net displacement of +2 m indicates a positive movement to the right, while the total distance traveled is 2 m back plus 2 m forward, resulting in 4 m.) In contrast, if you were to start at 2 m, move to 0 m, then to 4 m, delta x would still be +2 m, but the distance traveled would increase to 6 m (2 m back to 0 m and 4 m forward to 4 m). In both scenarios, the key distinction between displacement and distance becomes apparent, as displacement considers only the shortest path between start and finish points, while distance accounts for the entire path traveled.
Magnitude of displacement: |\Delta x| = 2\text{ m} (to the right).
Example 2: initial at 4 m, final at 2 m.
\Delta x = 2\text{ m} - 4\text{ m} = -2\text{ m}
Magnitude: |\Delta x| = 2\text{ m} (to the left).
Example 3 (three-step path): 2 m → 4 m → 0 m → 4 m
Segment displacements: Delta x1 = 4-2 = +2{ m}, \Delta x2 = 0-4 = -4{ m}, \Delta x_3 = 4-0 = +4{ m}
Total displacement: Delta x = \Delta x1 + \Delta x2 + \Delta x_3 = 2 - 4 + 4 = +2{ m}
Distance traveled: D = |+2| + |-4| + |+4| = 2 + 4 + 4 = 10\text{ m}
Time, Motion, and Time Intervals
Time is a scalar quantity (no direction).
Time interval (elapsed time) is:
Delta t = tf - ti Where tf is the final time and ti is the initial time, providing a measure of the duration of an event or motion in one dimension.
Most problems take the initial time as t_i = 0\,\text{ s} and report the final time as a positive value.
For motion with known displacement and time, you can compute average velocity (see below).
Notation and Conventions in 1D Kinematics
Position can be denoted as x, with initial value xi (or x0 }) and final value xfThe average velocity can be calculated using the formula:
v_{avg} = \frac{\Delta x}{\Delta t} = \frac{xf - x0}{tf - ti}
where ( \Delta x ) is the change in position, and ( \Delta t ) is the change in time.The instantaneous velocity, on the other hand, is defined as the limit of the average velocity as the time interval approaches zero:
v = rac{dx}{dt}
which describes the velocity of an object at a specific moment in time. In summary, while average velocity provides an overall rate of displacement during a given time interval, instantaneous velocity offers a precise measurement of speed and direction at any instant. Understanding these concepts is crucial for analyzing motion accurately, as they lay the foundation for further explorations into two-dimensional motion and acceleration. Furthermore, distinguishing between these two types of velocity helps in comprehending more complex motion scenarios, such as those involving changing speeds or directions.Displacement is denoted by \Delta x (sometimes written as the change in position from initial to final).
Distance traveled is denoted by a path-length quantity such as D or sometimes by the symbol for distance traveled along the path.
Sign conventions:
Rightward (positive x) directions are considered positive.
Leftward (negative x) directions are considered negative.
In 1D problems, many explanations are given on a single axis (x-axis); two-dimensional motion is introduced only briefly and later, as noted in the transcript.
Worked Practice Problems (Key Practice from Transcript)
Practice A: Displacement with positive direction
Object moves from 2 m to 4 m:
Delta x = xf - xi = 4{ m} - 2{ m} = +2{ m}This indicates that the object has moved 2 meters in the positive direction.
Sign indicates motion to the right.
Practice B: Displacement with negative direction
Object moves from 4 m to 2 m:
\Delta x = 2\text{ m} - 4\text{ m} = -2\text{ m}
Sign indicates motion to the left.
Practice C: Multi-segment path
Path: 2 m → 4 m → 0 m → 4 m
Segment displacements: Delta x1 = +2{ m}, \Delta x2 = -4{ m}, \Delta x_3 = +4{ m}The total displacement for the multi-segment path is calculated as follows:
( \Delta x{total} = \Delta x1 + \Delta x2 + \Delta x3 = +2{ m} - 4{ m} + 4{ m} = +2{ m} ).
The total distance traveled along the path, however, is the sum of the absolute values of each segment displacement, given by:
Distance = |Delta x1| + |Delta x2| + |Delta x3| = 2{ m} + 4{ m} + 4{ m} = 10{ m}. Therefore, it's important to differentiate between displacement and distance traveled in kinematic analyses, as they provide different insights into the motion of an object.
Total displacement: \Delta x = +2 - 4 + 4 = +2\text{ m}
Distance traveled: D = |+2| + |-4| + |+4| = 2 + 4 + 4 = 10\text{ m}
Practice D: Understanding final position, initial position, and displacement
If an object starts at 2 m and ends at 4 m after several motions, the displacement is +2\text{ m} and the distance traveled is the sum of the absolute values of each segment.
Average Velocity (from the concluding idea)
If you know displacement and time, you can calculate average velocity:
\bar{v} = \frac{\Delta x}{\Delta t}
Here, \Delta x = xf - xi and \Delta t = tf - ti.
Note: Average velocity is a vector quantity (in 1D it has a sign indicating direction); average speed would be the magnitude of the average velocity (or the total distance divided by total time, depending on convention).
Quick Recap and Connections
Key distinctions to remember:
Position: where the object is relative to the origin.
Displacement: the signed change in position between two points; a vector.
Distance traveled: the total length of the path traveled; a scalar.
Time interval: elapsed time between two instants.
Scalars vs Vectors: some quantities have only magnitude (scalars), others have magnitude and direction (vectors).
Foundational ideas connect to later topics:
Coordinate systems and sign conventions underpin velocity, acceleration, and force analyses.
Understanding 1D motion provides the basis for exploring 2D and 3D motion, where vectors are essential in multiple directions.
Practical Implications and Real-World Relevance
The distinction between displacement and distance is critical in navigation, tracking, and physics problems where net change vs path length matters (e.g., road trips, sports tracking).
Sign conventions and reference frames are essential in engineering, construction, and physics simulations to ensure consistent measurements and analyses.
Time intervals and average velocity are foundational for understanding motion in everyday contexts (driving, running, cycling, etc.).
Quick Reference Formulas (LaTeX)
Displacement: Delta x = xf - xi
Magnitude of displacement: |\Delta x|
Distance traveled (path length across segments): D = \sumi |\Delta xi| where ( \Delta xi ) represents the individual displacements along each segment of the path.
Time interval: Delta t = tf - ti
Average velocity (1D): \bar{v} = \frac{\Delta x}{\Delta t}
Position notation examples: xi,\ xf,\ x0,\ x
f,\
xi: Initial position\
xf: Final position\
x0: Position at time t = 0\
x: Position at any given time x(t): Represents the position as a function of time, allowing us to analyze motion at specific intervals.
One-dimensional sign convention: positive to the right/up, negative to the left/down ; this helps establish a common framework for understanding the direction and magnitude of various kinematic quantities.
Velocity (v): The rate of change of position with respect to time, which can be calculated as v = (xf - x0) / Δt, where Δt is the change in time.
Acceleration (a): The rate of change of velocity with respect to time, defined mathematically as a = (vf - v0) / Δt, where vf represents the final velocity and v0 is the initial velocity. This measurement is essential for understanding how the speed of an object changes over time.
Motion diagram: A visual representation of an object's position and motion over time, typically used to illustrate how an object's position changes in relation to its velocity and acceleration.
Displacement (d): The change in position of an object, which is a vector quantity that takes into account both the magnitude and direction of the overall change from the initial to the final position.
In summary, these foundational concepts in kinematics help us analyze and predict an object's motion through space. In addition to these concepts, understanding the distinction between scalars and vectors is crucial, as scalars are quantities that have magnitude only, such as distance and speed, whereas vectors have both magnitude and direction, such as displacement and velocity. The interplay between these quantities allows us to accurately describe motion and analyze the dynamics involved.