One-Dimensional Kinematics Notes (Transcript Summary)

Kinematics: One-Dimensional Motion

Kinematics is the description of motion without explaining why it happens. In this unit, we start by describing how an object moves on a line (one-dimensional motion).
Key setup for 1D motion:
- Choose an axis (a line) and an origin. Position along that axis is denoted by a coordinate, typically x (you may also see y for vertical motion).
- The direction along the axis matters: positive and negative signs indicate direction.
- Units: we use the MKS system. Position units are meters (m); time units are seconds (s).
- Position is a vector in general, so direction matters. In 1D motion, that direction is encoded with the sign of the coordinate.
Very first concepts you should know:
- Position: the location with respect to an axis.
- Displacement: the change in position over a time interval; symbol is
 $\Delta x = xf - xi,$
 where
- (x_i) = initial position (at the start of the time interval)
- (x_f) = final position (at the end of the time interval)
- Displacement is a vector (has direction) in 1D, so it has a sign.
- Distance traveled is the magnitude of the path length traveled, i.e., a scalar. It does not care about direction.
Notation and definitions (common in physics):
- Position:
 $x = \text{position along the axis}$
- Initial and final positions:
 $xi,\; xf$
- Often one writes initial velocity as $v0$ or sometimes $vi$ depending on context.
- For averages and other quantities, bar notation is sometimes used in textbooks, but it can be ambiguous with vectors. In this course, subscripts are preferred for clarity (e.g., (x0), (xf), (\bar{x}) if used).
1D motion on a line:
- The particle (or cart) moves back and forth along the line. Positive direction is conventionally taken as the rightward/upward direction, and negative is leftward/downward.
- The math is simpler in 1D, so we start there and then extend to 2D or 3D by adding components.
Graphs and time:
- Often, position is treated as a function of time: (x(t)). This means we plot position on one axis and time on the other.
- In many introductory physics courses, time is plotted on the x-axis and position on the y-axis. In some cases, you may see the axes swapped depending on the convention or the specific problem.
- When using graphing tools (e.g., Desmos), you may need a tiny adjustment because math courses usually plot (t) vs. (x) with time on the x-axis; in physics, you’ll frequently interpret the graph as (x) vs. (t).
- It’s important to keep clear what axis represents time and what axis represents position to avoid misreading the plot.
Position notation details:
- When introducing a new concept, the instructor often gives the term, then the symbol, e.g.,
- Position: (x) (location along the axis)
- Displacement: (\Delta x) (change in position)
- Subscripts vs parentheses:
- You may see (xi) (initial) and (xf) (final), or (x0, xt) depending on the author. The important part is that they refer to the values at the start and end of the interval.
Worked example (illustrative numbers from the transcript): a cart on a track moves in two segments:
- Segment 1: +60 cm (to the right)
- Segment 2: -40 cm (to the left)
- Displacement (net change in position):
 $\Delta x = (+60\ \text{cm}) + (-40\ \text{cm}) = +20\ \text{cm} = 0.20\ \text{m}.$
- Total distance traveled (magnitude of the path):
 $D = 60\ \text{cm} + 40\ \text{cm} = 100\ \text{cm} = 1.00\ \text{m}.$
- If we record positions at specific times (from the transcript):
- At (t = 1) s: (x = 2\ \text{m})
- At (t = 4) s: (x = 8\ \text{m})
- Displacement over this interval:
 $\Delta x = xf - xi = 8\ \text{m} - 2\ \text{m} = 6\ \text{m}.$
- Time interval:
 $\Delta t = 4\ \text{s} - 1\ \text{s} = 3\ \text{s}.$
- Average velocity over this interval:
 $\bar{v} = \frac{\Delta x}{\Delta t} = \frac{6\ \text{m}}{3\ \text{s}} = 2\ \text{m/s}.$
- Notes:
- The sign in displacement indicates direction; the average velocity here is positive, indicating overall motion in the positive direction during the interval.
- Distance traveled (1.00 m) is not the same as the displacement (0.20 m) for this two-segment path.
Distinguishing two related quantities:
- Displacement (a vector): net change in position, direction matters, can be less in magnitude than the total distance traveled.
- Distance (a scalar): total length of the path traveled, always nonnegative and does not include direction.
- Analogy: If you jog 6 km east and then 4 km west, your displacement is 2 km east, but your total distance jogged is 10 km.
Average quantities:
- Average velocity (a vector):
 $\bar{v} = \frac{\Delta x}{\Delta t} = \frac{xf - xi}{tf - ti}.$
- Average speed (a scalar):
 $\text{speed}_{avg} = \frac{D}{\Delta t},$
 where (D) is the total distance traveled.
- In practice, we often emphasize displacement for velocity calculations and distance for speed calculations.
Common notational conventions and cautions:
- Bar notation (e.g., (\bar{x}), (\bar{a})) can be used to denote averages in books, but it can be confusing with vectors (which use arrows above letters). The instructor prefers subscripts (e.g., (xi, xf, x_0)) for clarity in handwritten work.
- When writing, ensure you distinguish between a function notation (e.g., (x(t))) and a simple product (e.g., (x t)); context usually makes it clear, but be mindful to avoid misreading multiplication for a function.
- In lab-like examples, friction is often not ideal (e.g., a cart on a track is not perfectly frictionless; air resistance and track irregularities can cause drift), so the idealized equations are approximations.
Connections to broader physics:
- The discussion lays the foundation for graphs of position vs. time and for introducing calculus (derivatives) later in the course: velocity is the rate of change of position with respect to time, i.e.,
  $v(t) = \frac{dx}{dt}.$
- Later, one can extend to two-dimensional and three-dimensional motion by introducing components:
- In 2D: (\mathbf{r} = (x, y)) and (\mathbf{v} = (vx, vy)).
- In 3D: add the z-component: (\mathbf{r} = (x, y, z)) and (\mathbf{v} = (vx, vy, v_z)).
Quick summary of key formulas to memorize:
- Position: (x) (1D position along axis)
- Displacement: $\Delta x = xf - xi$
- Average velocity: $\bar{v} = \frac{\Delta x}{\Delta t} = \frac{xf - xi}{tf - ti}$
- Distance: distance traveled along the path (scalar) — not the same as displacement
- Average speed: $\text{speed}_{avg} = \frac{D}{\Delta t}$ where (D) is the total distance traveled
- Time axis conventions in graphs can vary; ensure you interpret the axes correctly in each graph
Practical takeaway:
- Start with clear definitions of 1D position, displacement, and distance.
- Use explicit subscripts for initial and final values when solving problems (e.g., (xi, xf, ti, tf)).
- Distinguish between vector quantities (which have direction) and scalar quantities (which do not).
- Practice converting between distance and displacement in simple two-segment motions to build intuition about how they differ.
If you’re preparing for assessments, you should be able to:
- Identify whether a given quantity is a vector or a scalar.
- Compute displacement from given initial and final positions.
- Compute distance traveled from a path description or from a table of successive displacements.
- Compute average velocity given a pair of positions and corresponding times, and interpret the sign of the result.
- Recognize the difference between velocity and speed and when to use each.
Real-world relevance:
- This framework is used in lab setups with carts on tracks, where you observe motion in a near-1D environment and measure positions at different times.
- In real experiments, factors like friction and air resistance are present, so ideal equations are approximations; recognizing this helps you interpret data correctly.