Kinematics and Dimensional Analysis Notes

Dimensional Consistency and Unit Conversions

• Principle: In physics, a full statement expressed by an equation must be dimensionally consistent; left and right sides must balance in both numbers and units.
• Practical recommendation: Practice converting units between different scales; this will be a recurring task throughout the course.
• Example: Converting miles per hour to meters per second.
  • Start with the basic definitions: 1 mile, 1 hour.
  • Substitute known values: 1 mile = 1610 m, 1 hour = 3600 s.
  • Therefore, $1\ \text{mph} = \frac{1\ \text{mile}}{1\ \text{hour}} = \frac{1610\ \text{m}}{3600\ \text{s}} \approx 0.447\ \text{m/s}.$
• Dimensional analysis exercise in the lecture: consider a quantity $z$ that depends on velocity, position, and time. If $z = \frac{2 v}{x t^2}$, what are the units of z?
  • Units: $[v] = [L][T]^{-1}, \ [x] = [L], \ [t] = [T].$
  • Therefore, $[z] = \frac{[v]}{[x][t]^2} = \frac{L T^{-1}}{L T^{2}} = T^{-3}.$
  • The constant 2 is dimensionless, so it acts as a multiplier.
• Conceptual takeaway: Dimensional consistency is a guide to check equations and to deduce possible units of unknown quantities.

Kinematics: Definitions and Key Quantities

• Kinematics: the science of motion and its description in space and time.
• Dynamics: the study of forces and what causes motion.
• Core quantities and relations:
  • Displacement, Δx: the change in position of an object along its path.
  • Distance: the path length traveled; can be greater than |Δx|.
  • Time interval, Δt: difference between final and initial times.
  • Position, x(t): the location along a chosen coordinate axis (here, a straight line, often along the y-axis in a one-dimensional setup).
• Important distinction:
  • Displacement is not the same as distance traveled; displacement is direction-aware (vector), while distance is the scalar path length.
  • Example: start at x0, return to x0: displacement Δx = 0, but distance traveled is nonzero.
• Straight-line motion setup:
  • Choose a simple one-dimensional coordinate system (along a straight line, e.g., the y-axis).
  • Track position x, time t, and determine velocity and acceleration from these data.

Uniform Motion: Data, Velocity, and Interpretation

• Uniform motion: motion with constant velocity; the velocity does not change with time.
• Data-taking approach (illustrative car on a straight road):
  • Snapshot 1: after 1 h, position x = 50 km.
  • Snapshot 2: after an additional 1 h (total 2 h), x = 100 km.
  • Snapshot 3: after 3 h, x = 150 km.
• From these data, compute average velocity on each interval:
  • For 1 h interval: $v_{avg} = \frac{Δx}{Δt} = \frac{50\ \text{km}}{1\ \text{h}} = 50\ \text{km/h}.$
  • For the next interval: $v_{avg} = \frac{100-50}{1} = 50\ \text{km/h}.$
  • For the next interval: $v_{avg} = \frac{150-100}{1} = 50\ \text{km/h}.$
• Conclusion: velocity is constant (uniform motion) in this example; average velocity equals the momentary velocity at any moment within these intervals.
• Notation: Often, the average velocity is denoted with an overbar: $\overline{v}$ or simply described as “average velocity.”
• Important note on interpretation:
  • When Δt is finite, the velocity computed is the average velocity over that time interval.
  • If Δt becomes smaller and smaller (approaching a point in time), the average velocity approaches the momentary velocity, which is the instantaneous rate of change of position.

Nonuniform Motion: Acceleration and Velocity Profiles

• Nonuniform motion: velocity changes with time (nonzero acceleration).
• Example data (nonuniform): consider a one-dimensional motion with data points (t, x):
  • (0 s, 0 m), (1 s, 1 m), (2 s, 4 m), (3 s, 9 m), (4 s, 16 m).
• Displacements per 1-second interval:
  • Δx from 0 to 1 s: 1 m.
  • Δx from 1 to 2 s: 3 m.
  • Δx from 2 to 3 s: 5 m.
  • Δx from 3 to 4 s: 7 m.
• Average velocities per interval: $v_{avg} = \frac{Δx}{Δt}$
  • 1 m/s, 3 m/s, 5 m/s, 7 m/s respectively.
• Observations:
  • The average velocity is increasing with time, indicating acceleration.
  • The acceleration can be computed as $a = \frac{Δv}{Δt}$; here, the change in velocity between successive intervals is 2 m/s per second, suggesting a constant acceleration of $a = 2\ \frac{\text{m}}{\text{s}^2}$ over these intervals.
• Moment velocity vs average velocity:
  • Moment velocity (instantaneous) is the limit of average velocity as Δt → 0, i.e., $v(t) = \frac{dx}{dt}$.
  • Graphically, average velocity over a finite interval is the slope of the secant line through the endpoints; moment velocity is the slope of the tangent line to the x(t) curve at a point.
• In this nonuniform example, the underlying motion is well described by x(t) = t^2 (in meters) for t in seconds, which yields:
  • Displacement: Δx = x(t2) − x(t1).
  • Velocity: v(t) = dx/dt = 2t (m/s).
  • Acceleration: a = dv/dt = 2 m/s^2 (constant in this example).
• Visual interpretation:
  • The average velocity sequence 1, 3, 5, 7 corresponds to the differences Δx = 1, 3, 5, 7 over successive unit time intervals.
  • The position vs time graph is a parabola when acceleration is constant (quadratic in t).
• Summary: Nonuniform motion involves nonzero acceleration, leading to changing velocity and a non-linear (parabolic) x(t) relationship when acceleration is constant.

Constant Acceleration: Key Equations of Motion

• If acceleration a is constant, velocity varies linearly with time and position follows a quadratic in time.
• Fundamental relations (for constant a):
  • Velocity as a function of time: $v(t) = v_0 + a t$
  • Position as a function of time: $x(t) = x<em>0 + v</em>0 t + \frac{1}{2} a t^2$
  • Velocity-squared relation (no explicit time): $v^2 = v<em>0^2 + 2 a (x - x</em>0)$
• Intuition and consistency:
  • A constant acceleration implies a straight line in the velocity-vs-time plot and a parabola in the position-vs-time plot.
  • The displacement over a time interval is the area under the velocity-time curve; for constant a this yields the quadratic form above.
• Example interpretation:
  • If starting from rest (v0 = 0) at x0, after time t with constant acceleration a, the displacement is $x(t) = x_0 + \frac{1}{2} a t^2$ and the velocity is $v(t) = a t$.
• Connecting to problem-solving:
  • In kinematics problems, once you know that acceleration is constant, you can apply the appropriate equation from the set above to solve for the desired quantity (displacement, time, velocity, etc.).

Graphical and Conceptual Insights: Secant vs Tangent, Parabolic Motion

• Visualization concepts:
  • Secant line: slope between two points on the x(t) curve; corresponds to the average velocity over the interval.
  • Tangent line: slope at a single point on the x(t) curve; corresponds to the momentary velocity v(t) = dx/dt.
• Under constant acceleration, the x(t) curve is a parabola in time; the velocity-time plot is a straight line (slope a).
• Recognition cue: When you see x ∝ t^2, you are in a constant-acceleration regime with a = constant.

Problem-Solving Strategy for Kinematics (Lecture’s Suggested Sequence)

• Step 1: Collect information about the motion.
  • Known: initial position x0, initial velocity v0, whether acceleration a is known and if it is constant.
  • Determine what is given and what is unknown (displacement, time, velocity, final position, etc.).
• Step 2: Determine the nature of the motion.
  • Is the acceleration constant? If yes, apply constant-acceleration equations.
  • If not, use the general definitions of velocity and acceleration and consider limits for moment velocity.
• Step 3: Choose the appropriate equations.
  • Use v = Δx/Δt for average velocity over a finite interval.
  • Use v = dx/dt for moment velocity (instantaneous).
  • Use x = x0 + v0 t + (1/2) a t^2 and v = v0 + a t for constant acceleration problems.
  • If needed, use the relation v^2 = v0^2 + 2 a (x - x0).
• Step 4: Compute and interpret the result.
  • Identify whether the answer is a displacement, time, velocity, or another quantity.
  • Check dimensional consistency and units at every step.
• Step 5: If multiple methods are possible, cross-check with alternate formulas to ensure consistency.

Quick Recap of Key Formulas and Concepts

• Fundamental definitions:
  • Average velocity: $\overline{v} = \frac{Δx}{Δt}$
  • Momentary (instantaneous) velocity: $v = \lim_{Δt \to 0} \frac{Δx}{Δt} = \frac{dx}{dt}$
• Displacement vs distance:
  • Displacement: $Δx = x - x_0$
  • Distance is the total path length traveled (always nonnegative).
• Uniform motion (constant velocity):
  • Velocity is constant; same velocity across all intervals.
• Nonuniform motion (variable velocity):
  • Acceleration a = Δv/Δt > 0 or < 0; velocity changes over time.
• Constant-acceleration kinematics:
  • $v(t) = v_0 + a t$
  • $x(t) = x<em>0 + v</em>0 t + \frac{1}{2} a t^2$
  • $v^2 = v<em>0^2 + 2 a (x - x</em>0)$
• Units and conversions (quick references):
  • SI base units: meters (m), seconds (s), kilograms (kg), etc.
  • Example conversion: $1\ \text{mph} = \frac{1\ \text{mile}}{1\ \text{hour}} = \frac{1610\ \text{m}}{3600\ \text{s}} \approx 0.447\ \text{m/s}.$
• Dimensional check example:
  • If $z = \frac{2 v}{x t^2}$, then $[z] = \frac{[v]}{[x][t]^2} = \frac{L T^{-1}}{L T^{2}} = T^{-3}.$
• Real-world relevance:
  • The careful handling of units and dimensions prevents errors in physics modeling and simulation.
  • The stepwise approach to kinematics builds a foundation for dynamics, work, and energy in later chapters.