AJ

Kinematics Vocabulary: Scalars, Vectors, Distance, Displacement, Speed, and Velocity

Scalars vs Vectors and Motion

  • Kinematics is the branch of physics that studies motion and how to describe it. The term comes from a root related to motion.
  • Two basic kinds of mathematical quantities used to describe motion:
    • Scalars: quantities with only magnitude (no direction).
    • Vectors: quantities with both magnitude and direction; require at least two numbers to specify (e.g., magnitude and direction, or components along axes).
  • Examples from the lecture:
    • Microwave on high for forty seconds: has a duration but no direction → scalar.
    • The speed limit of 60 miles per hour: has a magnitude but no direction → scalar.
    • A distance range like 30 feet: describes length but not direction → not a vector by itself.
    • Motion along a river (downriver) introduces a definite direction → a vector situation.
  • In 1D, a vector reduces to a signed quantity along the line (direction is indicated by the sign).
  • In 2D or 3D, vectors have components along multiple axes; this will be covered in later chapters.
  • Absolute position is not the only or most important quantity in physics; what matters are changes in position (displacements) and how they evolve in time.

Coordinate systems and invariance of displacement

  • Coordinates are a matter of choice; you can pick any origin and axes you like.
  • If you shift the coordinate origin by adding a constant to all positions, the physical displacement does not change.
    • If x' = x + c, then the displacement between two events is invariant:

      \Delta x' = xf' - xi' = (xf + c) - (xi + c) = xf - xi = \Delta x.
  • Because of this, physics often focuses on changes in position (displacement) rather than absolute positions.

Distance vs. Displacement (1D intuition)

  • Distance (path length) is the total length traveled along the path and is a scalar.
  • Displacement is the net change in position: the vector from the initial position to the final position.
  • In one dimension (monotonic motion along a line), distance traveled can equal the absolute value of displacement:
    s = |\Delta x| = |xf - xi|.
  • Non-monotonic motion (changing direction) makes distance differ from |\Delta x| since distance sums along the actual path.
  • Example from the transcript (1D track):
    • Let initial position be $xa = -1$ and a later position be $xc = 5.5$. Then
      \Delta x{ac} = xc - x_a = 5.5 - (-1) = 6.5.
    • If this motion is in a single direction, the distance traveled from a to c is $s{ac} = |\Delta x{ac}| = 6.5$.
    • The transcript also notes a concrete set of numbers for the positions (e.g., a = -1, b = 2, c = 5.5, d = 2) to illustrate how changing the coordinate origin does not change the displacement in value.

Time and the change in time

  • Motion descriptions require time as a changing quantity.
  • Time is denoted by $t$, and a change in time is denoted $\Delta t = tf - ti$.
  • In many problems, you can set the starting time $t_0 = 0$ and measure elapsed time relative to that moment.
  • When plotting motion, one common convention is to place time on the horizontal axis and position on the vertical axis (the lecture mentions this plotting choice).

Speed vs. Velocity

  • Speed is a scalar: it is the rate of motion without regard to direction.
    • Definition: v = \frac{\text{distance}}{\text{time}} = \frac{\Delta s}{\Delta t}
    • Common symbol: $v$ for speed (magnitude of velocity).
  • Velocity is a vector: it includes both speed and direction.
    • In 1D, velocity has a sign indicating direction along the line.
    • In higher dimensions, velocity is a vector with components (e.g., $vx$, $vy$, $v_z$).
  • Relationship between displacement and average velocity (1D example):
    • From A to C, the displacement is $\Delta x{ac} = 6.5$ and the time interval is $\Delta t{ac} = 13\ \text{s}$, so the average velocity is
      v{\text{avg,ac}} = \frac{\Delta x{ac}}{\Delta t_{ac}} = \frac{6.5}{13} = 0.5.
  • The lecture clarifies that average velocity uses the overall displacement divided by total time, while instantaneous velocity is the velocity at a single moment (not necessarily equal to the average).

Instantaneous velocity and the tangent concept

  • Instantaneous velocity is the velocity at a specific moment, which corresponds to the slope of the position-versus-time curve at that moment.
  • If the position as a function of time $x(t)$ is changing, the instantaneous velocity at a point is:
    v{inst} = \lim{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}.
  • In practice (as described in the lecture), you can approximate instantaneous velocity by the slope of the tangent to the $x$ vs $t$ graph at the point of interest.
  • Example given: the tangent slope at point $b$ is approximately $0.7338$ (units depend on the units of $x$ and $t$).
  • The lecture emphasizes that, for this course, calculus-based computation of derivatives is not required to compute instantaneous velocity, but the concept of a tangent slope is essential for understanding instantaneous rates of change.

Graphical interpretation of velocity on a motion plot

  • A position-versus-time graph $x(t)$ has:
    • Horizontal axis: time $t$ (as used in the lecture’s plotting convention).
    • Vertical axis: position $x$.
  • The slope of the graph at any point equals the instantaneous velocity at that time:
    v(t) = \frac{dx}{dt} = \text{slope of the tangent at time } t.
  • The average velocity over an interval is the slope of the straight line connecting the endpoints of the interval on the $x$ vs $t$ graph.

Key takeaways and conceptual distinctions

  • Distance vs. displacement:
    • Distance is the total length traveled (scalar).
    • Displacement is the net change in position (vector, with sign in 1D).
  • Scalars vs. vectors:
    • Scalars have magnitude only; vectors have magnitude and direction.
    • In 1D, vectors reduce to signed quantities; in higher dimensions, direction matters along multiple axes.
  • Coordinate choice:
    • The physics of motion depends on changes in position, not on the absolute value of position.
    • Shifting the origin does not change displacement or velocity (since velocity relates to the rate of change, which is origin-independent up to a constant offset in position).
  • Time and motion:
    • Motion descriptions require time; you often start at $t_0 = 0$ for convenience.
    • Time is the independent variable in $x(t)$, and velocity is the rate of change of $x$ with respect to $t$.
  • Instantaneous velocity vs. average velocity:
    • Average velocity uses a finite interval: v_{avg} = \frac{\Delta x}{\Delta t}.
    • Instantaneous velocity is a limit (or slope of the tangent): v{inst} = \frac{dx}{dt} = \lim{\Delta t\to 0} \frac{\Delta x}{\Delta t}.
  • Practical approach in introductory kinematics:
    • You can use the tangent slope on a graph to estimate instantaneous velocity without doing full calculus, though the underlying definition is a limit that requires derivatives.

Practical and cross-cutting points

  • Relative motion and frames of reference: choosing different coordinate origins or axes does not affect physically measurable quantities like displacement, velocity, or acceleration.
  • Real-world relevance: speed limits, durations (e.g., 13 s), distances (e.g., 6.5 units), and directions all play into predicting motion and planning trajectories.
  • This foundational material sets up further topics such as acceleration, motion in two dimensions, and more advanced kinematic concepts in subsequent chapters.