Physics 1 Notes: Units, Dimensional Analysis, Trig, Vectors, and Kinematics (1D & 2D)

SI Units, Base Quantities, and Dimension Analysis

SI base units (ground level building blocks):
- Length: meter, units $ext{m}$
- Time: second, units $ext{s}$
- Mass: kilogram, units $ext{kg}$
- Correction note: transcript says “mass in kilohertz” which is a mistake; mass is in kilograms.
Derived units of common interest:
- Velocity: $ext{m/s}$
- Acceleration: $ext{m/s}^2$
- Force: Newton, $ext{N} = ext{kg}\, ext{m}/ ext{s}^2$
- Energy (work): Joule, $ext{J} = ext{N}\, ext{m} = ext{kg}\, ext{m}^2/ ext{s}^2$
Unit conversions (dimensional analysis): start with the unit you have and multiply by a conversion factor so that units cancel appropriately.
- Example setup: to convert from meters to centimeters, use the factor $1 ext{ m}=100 ext{ cm}$ , place the unit you have on the bottom and the unit you want on the top, add a factor of 1 to keep the conversion neutral.
- Quick check: meters canceled out, leaving centimeters.
Quick practical conversion example (km to miles):
- Given: $1 ext{ mile} = 1609.34 ext{ m}$ and you have $10 ext{ km} = 10{,}000 ext{ m}$ .
- Calculation:
- ext{Miles} = rac{10{,}000}{1609.34} \, ext{miles}
- ≈ $6.2137 ext{ miles}$
- Report with appropriate precision: typically three significant figures → $6.21 ext{ miles}$ .
Dimensional consistency check (example):
- Consider the equation $x = x0 + v0 t + frac{1}{2} a t^2$
- Dimensions:
- $[x] = [x_0] = L\,,$
- $[v_0 t] = [L/T] \cdot T = L\,$
- $[a t^2] = [L/T^2] \cdot T^2 = L\,$
- All terms have dimension $L$ ; thus the equation is dimensionally consistent.
Common pitfalls in dimensional analysis:
- When converting, cancel units correctly as you place factors diagonally.
- Always compare dimensions, not just numbers, to catch mistakes (e.g., a velocity vs. an acceleration).

Scientific Notation and Significant Figures (sig figs)

Scientific notation basics:
- A value is written as $a imes 10^n$ with 1 \,\le\, |a| < 10 and integer $n$ .
- Example: $0.00123 = 1.23 \times 10^{-3}$ .
Significant figures rules (non-exact measurements):
- Nonzero digits are always significant.
- Zeros between nonzero digits are significant.
- Leading zeros are not significant.
- Trailing zeros are significant if a decimal point is present; otherwise they may not be.
- Exact counts have infinite sig figs (e.g., defined quantities or counted items).
Multiplication and division with sig figs:
- The result should have as many sig figs as the factor with the fewest sig figs.
Addition and subtraction with sig figs:
- The result should have the decimal place of the least precise measurement (fewest decimal places), not the fewest sig figs.
Uncertainty terminology:
- Absolute uncertainty: $\Delta x$ — the last digit in the measurement.
- Relative uncertainty:
- $\% \,\text{relative uncertainty} = \left(\frac{\Delta x}{x}\right) \times 100\%$
Practical notes on uncertainty:
- Real measurements have uncertainty; for example, a ruler with resolution 0.1 mm implies a measurement uncertain by about ±0.1 mm.
- If a quantity is an exact count, it has infinite sig figs; it does not limit the precision of the result.
Example (sig figs in operation):
- If you have two measurements, $3.1$ (2 sig figs) and $7.2$ (2 sig figs) added, the result should be reported with the least decimal places; for this case, tenths place, so the sum is reported with one decimal place.

Trigonometry, SOHCAHTOA, and Angles

SOHCAHTOA mnemonic (right triangles only):
- Sine: $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$
- Cosine: $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
- Tangent: $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$
Relationship to vectors (magnitude and components):
- Given a vector of magnitude $|\mathbf{v}|$ at angle $\theta$ (from the +x axis):
- $v_x = |\mathbf{v}| \cos\theta$
- $v_y = |\mathbf{v}| \sin\theta$
Angle conventions and quadrant checks:
- When you know components ( $vx, vy$ ) but not the angle, you can compute:
- $\theta = \arctan\left(\frac{vy}{vx}\right)$ with quadrant correction, or
- $\theta = \arccos\left(\frac{v_x}{|\mathbf{v}|}\right)$ , or
- $\theta = \arcsin\left(\frac{v_y}{|\mathbf{v}|}\right)$
- Example (given components: $vx=-2, vy=6$ ):
- Magnitude $|\mathbf{v}| = \sqrt{(-2)^2 + 6^2} = \sqrt{40} \approx 6.32$
- $\theta = \arctan2(6,-2) \approx 108.4^{\circ}$ (Quadrant II)
Degrees vs radians:
- Full circle: $360^{\circ} = 2\pi \,\text{rad}$
- $\pi \,\text{rad} = 180^{\circ}$
- Common conversions: $0^{\circ} = 0\,\text{rad}, \ 90^{\circ} = \tfrac{\pi}{2}\,\text{rad}, \ 180^{\circ} = \pi\,\text{rad}$

Vectors: Magnitude, Components, and Unit Vectors

A vector has both magnitude and direction; a scalar has magnitude only.
Component form: $\mathbf{v} = vx \hat{i} + vy \hat{j}$
Magnitude and direction:
- $|\mathbf{v}| = \sqrt{vx^2 + vy^2}$
- If the angle is known: $vx = |\mathbf{v}| \cos\theta, \ vy = |\mathbf{v}| \sin\theta$
Unit vectors:
- Along x: $\hat{i}$ , along y: $\hat{j}$
- Any vector can be written as a combination of unit vectors: $\mathbf{v} = vx \hat{i} + vy \hat{j}$
Example: add two vectors
- Given, for instance, $\mathbf{a} = (3,4)$ and a second vector $\mathbf{b} = (-5,2)$ , the resultant is
- $\mathbf{r} = \mathbf{a} + \mathbf{b} = (-2, 6)$
- Magnitude: $|\mathbf{r}| = \sqrt{(-2)^2 + 6^2} = \sqrt{40} \approx 6.32$
- Direction: $\theta = \arctan2(6,-2) \approx 108.4^{\circ}$ (Quadrant II)
Practical note on signs: the quadrant of the resultant determines the sign of its components; you can use arctan2 to handle signs consistently.

Kinematics in One Dimension (Motion in 1D)

Displacement: $\Delta s = s - s_0$
Velocity: $\mathbf{v} = \frac{\Delta s}{\Delta t}$
- Velocity is a vector (has direction); speed is its magnitude (scalar).
- Instantaneous velocity is the velocity at a single moment; average velocity is over an interval.
Acceleration: $\mathbf{a} = \frac{\Delta \mathbf{v}}{\Delta t}$
Graphical interpretations:
- S vs t: slope = velocity; velocity vs t: slope = acceleration; area under v vs t: displacement.
Key takeaway: velocity and acceleration are derivatives of position with respect to time; speed and velocity differ by whether direction is included.

Constant Acceleration: The Kinematic Equations (One Dimension)

Assumption: acceleration is constant during the motion.
The three core equations (with initial values denoted by subscript 0):
- $v = v_0 + a\,t$
- $s = s0 + v0\,t + \tfrac{1}{2} a t^2$
- $v^2 = v0^2 + 2 a (s - s0)$
Variables involved: final velocity $v$ , initial velocity $v0$ , acceleration $a$ , time $t$ , displacement $s$ , initial position $s0$ .
How to pick equations:
- Use the equation that matches the known quantities you’re given (and what you’re asked to find).
- If acceleration is constant, these three cover most problems; if acceleration isn’t constant, these don’t apply.
Quick worked example (car from rest with $a=3\,\text{m/s}^2$ for $t=4\,\text{s}$ ):
- Initial velocity $v0 = 0$ , final velocity $v = v0 + a t = 0 + 3(4) = 12\,\text{m/s}$
- Displacement
 $s = s0 + v0 t + \tfrac{1}{2} a t^2 = 0 + 0\cdot 4 + \tfrac{1}{2} (3) (4)^2 = 24\,\text{m}$
Note on units: the velocity unit is meters per second (m/s); acceleration unit is meters per second squared (m/s^2).

Free Fall and Projectile Motion (Motion in 2D, gravity)

Free fall basics:
- Only force is gravity; acceleration is constant: $a_y = -g,\quad g\approx 9.81\, \text{m/s}^2$ (sign depends on chosen convention; you must stay consistent).
- Horizontal acceleration $a_x = 0$ (if no air resistance).
Upward throw example: initial vertical velocity $v_{0y} = 10\,\text{m/s}$ from ground.
- Time to reach max height: $t{\text{top}} = \frac{v{0y}}{g} \approx \frac{10}{9.81} \approx 1.02\ \text{s}$
- Maximum height: $y{\text{max}} = \frac{v{0y}^2}{2g} \approx \frac{100}{19.62} \approx 5.10\ \text{m}$
- Time of flight (up and down to same level): $T = \frac{2 v_{0y}}{g} \approx \frac{20}{9.81} \approx 2.04\ \text{s}$
Projectile motion (two dimensions): decompose into components
- Initial velocity components: $v{0x} = v0 \cos\theta, \ v{0y} = v0 \sin\theta$
- Equations of motion (assuming origin at launch point):
- Horizontal: $x(t) = x0 + v{0x} t$ , with $a_x = 0$
- Vertical: $y(t) = y0 + v{0y} t - \tfrac{1}{2} g t^2$ , with $a_y = -g$
Independence of axes: motion in x does not affect motion in y and vice versa; solve each axis separately and combine.
If initial velocity components are given, angle can be recovered from components:
- Magnitude: $v0 = \sqrt{v{0x}^2 + v_{0y}^2}$
- Angle: $\theta = \arctan\left(\frac{v{0y}}{v{0x}}\right)$ (with quadrant adjustment as needed).
Key projectile quantities and shortcuts:
- Time to max height: $t{\text{top}} = \frac{v{0y}}{g}$
- Max height: $y{\text{max}} = \frac{v{0y}^2}{2g}$
- Time of flight for symmetric launch and landing: $T = \frac{2 v_{0y}}{g}$
- Range (horizontal distance) often derived from $x(T) = v_{0x} T$ when landing height equals launch height.
Practical sign convention note: you may take upward as positive or downward as positive; keep your sign convention consistent throughout a problem.

Two-Dimensional Motion: Components and Magnitude

Splitting 2D motion into x- and y- components:
- Displacement: $\Delta x = v{0x} t$ (if $ax=0$ ) and $\Delta y = v_{0y} t - \tfrac{1}{2} g t^2$
- Velocity components: $vx(t) = v{0x}$ (constant if $ax=0$ ), $vy(t) = v_{0y} - g t$
Resultant velocity magnitude at time t:
- $|\,\mathbf{v}(t)\,| = \sqrt{vx(t)^2 + vy(t)^2}$
Direction of velocity: $\theta(t) = \tan^{-1}\left(\frac{vy(t)}{vx(t)}\right)$

Relative Velocity

Relative velocity depends on the frame of reference:
- General rule: $\mathbf{v}{A/C} = \mathbf{v}{A/B} + \mathbf{v}_{B/C}$
This lets you transform velocities between frames (A relative to C) by adding the velocity of B relative to C to the velocity of A relative to B.

Dimensional Analysis in Practice (Quick Recap from Lecture)

Use dimension to check equations and practice problems that combine displacement, velocity, and acceleration.
Example equation check (dimensionally consistent):
- $x = x0 + v0 t + \tfrac{1}{2} a t^2$
- Units: $[x] = L, [x0] = L, [v0 t] = (L/T) \cdot T = L, [a t^2] = (L/T^2) \cdot T^2 = L$

Worked Example: Two Vectors and The Resultant

Given vectors: $\mathbf{a} = (3,4)$ and $\mathbf{b} = (-5,2)$ (components along x and y)
Resultant: $\mathbf{r} = \mathbf{a} + \mathbf{b} = (3-5,\ 4+2) = (-2,6)$
Magnitude: $|\mathbf{r}| = \sqrt{(-2)^2 + 6^2} = \sqrt{40} \approx 6.32$
Direction: $\theta = \tan^{-1}\left(\frac{ry}{rx}\right) = \tan^{-1}\left(\frac{6}{-2}\right) = \tan^{-1}(-3) \approx -71.6^{\circ}$
- Correct for quadrant: since rx
Alternative method: use $\theta = \cos^{-1}\left(\frac{rx}{|\mathbf{r}|}\right)$ or $\theta = \sin^{-1}\left(\frac{ry}{|\mathbf{r}|}\right)$ with quadrant checks.
Note on notation: sometimes you’ll see the vector written as $\mathbf{r} = |\mathbf{r}| (\cos\theta \hat{i} + \sin\theta \hat{j})$ .

Quick Reference: Key Formulas (Summary)

Displacement (1D): $\Delta s = s - s_0$ ; Velocity: $\mathbf{v} = \frac{\Delta s}{\Delta t}$ ; Acceleration: $\mathbf{a} = \frac{\Delta \mathbf{v}}{\Delta t}$
Kinematic equations (constant acceleration):
- $v = v_0 + a t$
- $s = s0 + v0 t + \tfrac{1}{2} a t^2$
- $v^2 = v0^2 + 2 a (s - s0)$
Projectile motion (components):
- $x(t) = x0 + v{0x} t$ , with $a_x = 0$
- $y(t) = y0 + v{0y} t - \tfrac{1}{2} g t^2$ , with $a_y = -g$
- $v{0x} = v0 \cos\theta, \ v{0y} = v0 \sin\theta$
Max height, flight time, and range (for vertical launch):
- $t{\text{top}} = \frac{v{0y}}{g}$
- $y{\text{max}} = \frac{v{0y}^2}{2g}$
- $T = \frac{2 v_{0y}}{g}$
Vector relations:
- $|\mathbf{v}| = \sqrt{vx^2 + vy^2}$
- $vx = |\mathbf{v}| \cos\theta, \ vy = |\mathbf{v}| \sin\theta$
- $\mathbf{v} = vx \hat{i} + vy \hat{j}$
- Use $\theta = \tan^{-12}(vy, vx)$ to get the correct quadrant.
Relative velocity:
- $\mathbf{v}{A/C} = \mathbf{v}{A/B} + \mathbf{v}_{B/C}$
Units and uncertainty:
- Absolute uncertainty: $\Delta x$
- Relative uncertainty: $\left(\frac{\Delta x}{x}\right) \times 100\%$
- Three sig figs rule for reporting; least precise decimal place rule for addition/subtraction.

Worked Practice Problems Suggested for Self-Study

Dimensional analysis check: verify the equation $x = x0 + v0 t + \tfrac{1}{2} a t^2$ is dimensionally consistent.
1D constant acceleration problem: a car starts from rest with $a = 3\,\text{m/s}^2$ for $t=4\,\text{s}$ . Find $v$ and $s$ .
Free-fall problem: a ball is thrown upward with $v_0 = 10\,\text{m/s}$ . Find time to top, max height, and total time of flight (take $g = 9.81\,\text{m/s}^2$ ).
Projectile example: given initial speed and angle, decompose into components and solve for position after a given time or for time to reach a given height.
Vector addition: given two 2D vectors, compute resultant vector, its magnitude, and direction; verify quadrant of the resultant and adjust angle accordingly.
Relative velocity in different frames: practice transforming a velocity from one frame to another using the relative velocity formula above.

Quick Summary of Chapter Flow (Contextual Connection)

Section 1: Units, dimensions, and prefixes; dimension analysis for consistency.
Section 2: Scientific notation and sig figs; rules for multiplication/division and addition/subtraction; measurement uncertainty.
Section 3: Trigonometry basics (SOHCAHTOA); unit circle and angle conventions; vector components and quadrants.
Section 4: Vectors; magnitude, components, unit vectors; vector addition and resultant.
Section 5: Kinematics in one dimension; displacement, velocity, acceleration; graphical interpretations.
Section 6: Constant acceleration kinematics; the three core equations and their applicability.
Section 7: Free fall and projectile motion; 2D motion; independence of axes; horizontal and vertical components;
time-to-height, range, and flight time shortcuts.
Section 8: Multi-dimensional motion and relative velocity concepts; formula for relative velocity in different frames.
Ethical, philosophical, or practical implications discussed in the lecture:
- Measurement uncertainty as a fundamental part of experimental science; all measurements have a limit to precision defined by instrument resolution (the last digit is the uncertainty).
- The importance of dimensional consistency as a basic check on physical equations, preventing misapplication of formulas to real problems.
- In experimental practice, exact counts are considered to have infinite sig figs, which influences data reporting and error analysis.
Note on numerical values observed in the lecture:
- Some numbers in the transcript (e.g., certain trig evaluations or unit labels) contain typographical or arithmetic inaccuracies. The notes above present the standard, correct forms and typical numerical results
 (e.g., for a 12 m/s vector at 35°, components are $vx = 12\cos 35^{\circ} \approx 9.83\,\text{m/s}, \ vy = 12\sin 35^{\circ} \approx 6.89\,\text{m/s}$ ).