Physics with Calculus: Units, Physical Quantities, Vectors, and Motion (Chapters 1–5) – Mini-Textbook Study Notes

Units, Physical Quantities, and Vectors

The Nature of Physics

Physics is an experimental science that seeks to describe the fundamental laws governing the universe. Physical theories are developed through observation, experimentation, and logical reasoning. Theories are always subject to revision based on new evidence.

• Physical Theory: An explanation of natural phenomena based on observation and accepted principles.

• Physical Law: A well-established theory widely used and supported by evidence.

• Idealized Model: A simplified version of a physical system that captures essential features while ignoring minor details.

Solving Physics Problems

Problem-solving in physics involves a systematic approach:

1. Identify relevant concepts and target variables.

2. Set Up the problem by choosing equations and drawing diagrams.

3. Execute the solution by performing calculations.

4. Evaluate the answer for reasonableness and accuracy.

Standards and Units

Physical quantities are measured using standardized units. The International System of Units (SI) is the standard in science.

• Fundamental SI Units: meter (m) for length, kilogram (kg) for mass, second (s) for time.

• Operational Definition: Defines a quantity by describing how to measure it (e.g., using a ruler for length).

• Derived Units: Formed from combinations of fundamental units (e.g., speed in m/s).

Unit Prefixes

Prefix	Symbol	Factor
kilo-	k	103
centi-	c	10-2
milli-	m	10-3
micro-	μ	10-6
nano-	n	10-9

Using and Converting Units

• Always include units in calculations.

• Use unit multipliers to convert between units (e.g., $1\ \text{min} = 60\ \text{s}$).

• Equations must be dimensionally consistent.

Uncertainty and Significant Figures

• Uncertainty: All measurements have some error; report as $x \pm \Delta x$.

• Significant Figures: The number of meaningful digits in a measurement.

• Rules: For multiplication/division, the result has as many significant figures as the factor with the fewest. For addition/subtraction, the result is limited by the least precise decimal place.

Example:

• $3.1416 \times 2.34 \times 0.58 = 4.3$ (two significant figures)

• $123.62 + 8.9 = 132.5$ (tenths place)

Estimates and Orders of Magnitude

• Order-of-Magnitude Estimate: A rough calculation, often accurate within a factor of 10.

• Useful for checking reasonableness of answers.

Vectors and Vector Addition

• Scalar: Quantity with magnitude only (e.g., mass, temperature).

• Vector: Quantity with both magnitude and direction (e.g., displacement, velocity, force).

• Vectors are added graphically (head-to-tail) or by components.

• Vector Addition: $\vec{C} = \vec{A} + \vec{B}$

• Vector Subtraction: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

• Multiplying by a Scalar: Changes magnitude, not direction (unless scalar is negative).

Components of Vectors

• Any vector in a plane can be written as $\vec{A} = A_x \hat{\imath} + A_y \hat{\jmath}$

• $A_x = A \cos \theta$, $A_y = A \sin \theta$ (if $\theta$ is measured from the $+x$-axis)

• Magnitude: $A = \sqrt{A_x^2 + A_y^2}$

• Direction: $\tan \alpha = \frac{A_y}{A_x}$

Unit Vectors

• Unit vectors $\hat{\imath}$, $\hat{\jmath}$, $\hat{k}$ point in the $x$, $y$, $z$ directions, respectively, and have magnitude 1.

• Any vector: $\vec{A} = A_x \hat{\imath} + A_y \hat{\jmath} + A_z \hat{k}$

Products of Vectors

• Scalar (Dot) Product: $\vec{A} \cdot \vec{B} = AB \cos \phi = A_x B_x + A_y B_y + A_z B_z$

• Vector (Cross) Product: $\vec{A} \times \vec{B}$ is a vector perpendicular to both $\vec{A}$ and $\vec{B}$, with magnitude $AB \sin \phi$ and direction given by the right-hand rule.

• Components: $C_x = A_y B_z - A_z B_y$, $C_y = A_z B_x - A_x B_z$, $C_z = A_x B_y - A_y B_x$

Motion Along a Straight Line

Displacement, Time, and Average Velocity

• Displacement: $\Delta x = x_2 - x_1$

• Average Velocity: $v_{\text{av}-x} = \frac{\Delta x}{\Delta t}$

• Direction matters: positive for motion in $+x$, negative for $-x$.

Instantaneous Velocity

• $v_x = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$

• Instantaneous speed is the magnitude of velocity.

Average and Instantaneous Acceleration

• Average Acceleration: $a_{\text{av}-x} = \frac{\Delta v_x}{\Delta t}$

• Instantaneous Acceleration: $a_x = \frac{dv_x}{dt}$

Motion with Constant Acceleration

• Equations of motion (for constant $a_x$):

• $v_x = v_{0x} + a_x t$

• $x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$

• $v_x^2 = v_{0x}^2 + 2 a_x (x - x_0)$

• $x - x_0 = \frac{1}{2}(v_{0x} + v_x)t$

Freely Falling Objects

• Acceleration due to gravity: $g \approx 9.80\ \text{m/s}^2$ (downward)

• For vertical motion: $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

• At maximum height, $v_y = 0$

Velocity and Position by Integration

• If $a_x$ is not constant:

• $v_x = v_{0x} + \int_0^t a_x dt$

• $x = x_0 + \int_0^t v_x dt$

Motion in Two or Three Dimensions

Position and Velocity Vectors

• Position: $\vec{r} = x \hat{\imath} + y \hat{\jmath} + z \hat{k}$

• Average velocity: $\vec{v}_{\text{av}} = \frac{\Delta \vec{r}}{\Delta t}$

• Instantaneous velocity: $\vec{v} = \frac{d\vec{r}}{dt}$

The Acceleration Vector

• Average acceleration: $\vec{a}_{\text{av}} = \frac{\Delta \vec{v}}{\Delta t}$

• Instantaneous acceleration: $\vec{a} = \frac{d\vec{v}}{dt}$

• Components: $a_x = \frac{dv_x}{dt}$, $a_y = \frac{dv_y}{dt}$, $a_z = \frac{dv_z}{dt}$

• Acceleration can be decomposed into components parallel and perpendicular to velocity.

Projectile Motion

• Projectile motion is two-dimensional, with $a_x = 0$, $a_y = -g$.

• Equations (assuming launch from origin):

• $x = v_0 \cos \theta_0 t$

• $y = v_0 \sin \theta_0 t - \frac{1}{2} g t^2$

• Trajectory is a parabola: $y = (\tan \theta_0) x - \frac{g}{2 v_0^2 \cos^2 \theta_0} x^2$

• Maximum height: $h = \frac{v_0^2 \sin^2 \theta_0}{2g}$

• Range (if landing height = launch height): $R = \frac{v_0^2 \sin 2\theta_0}{g}$

Motion in a Circle

• Uniform circular motion: speed constant, direction changes.

• Radial (centripetal) acceleration: $a_{\text{rad}} = \frac{v^2}{R}$

• Period: $T = \frac{2\pi R}{v}$

• Radial acceleration in terms of period: $a_{\text{rad}} = \frac{4\pi^2 R}{T^2}$

• Nonuniform circular motion: also has tangential acceleration $a_{\text{tan}} = \frac{dv}{dt}$

Relative Velocity

• Relative velocity in one dimension: $v_{P>A} = v_{P>B} + v_{B>A}$

• In two or three dimensions: $\vec{v}_{P>A} = \vec{v}_{P>B} + \vec{v}_{B>A}$

• Velocities add as vectors.

Newton's Laws of Motion

Force and Interactions

• Force: A push or pull, a vector quantity, always the result of an interaction between two objects.

• Contact Forces: Normal force (perpendicular to surface), friction force (parallel to surface), tension force (pulling via rope or cable).

• Long-range Forces: Gravity, electromagnetic forces.

• Superposition Principle: The net force is the vector sum of all individual forces: $\vec{R} = \sum \vec{F}$

Newton's First Law (Law of Inertia)

• If $\sum \vec{F} = 0$, an object at rest remains at rest, and an object in motion continues with constant velocity.

• Inertia: The tendency of an object to resist changes in its motion.

• Equilibrium: Zero net force, zero acceleration.

• Inertial Frame: A reference frame in which Newton's first law holds (not accelerating).

Newton's Second Law

• The net external force on an object equals mass times acceleration: $\sum \vec{F} = m \vec{a}$

• Component form: $\sum F_x = m a_x$, $\sum F_y = m a_y$, $\sum F_z = m a_z$

• SI unit of force: newton (N), $1\ \text{N} = 1\ \text{kg} \cdot \text{m}/\text{s}^2$

• Only external forces affect an object's motion; internal forces cancel.

Mass and Weight

• Mass: A measure of an object's inertia; does not change with location.

• Weight: The gravitational force on an object: $w = mg$

• Weight varies with $g$ (e.g., on the Moon, $g$ is smaller).

Newton's Third Law

• If object A exerts a force on object B, then B exerts an equal and opposite force on A: $\vec{F}_{A\ on\ B} = -\vec{F}_{B\ on\ A}$

• Action–reaction pairs act on different objects.

Free-Body Diagrams

• Draw all external forces acting on the object of interest.

• Do not include forces the object exerts on others, or internal forces.

• Forces are represented as arrows pointing in the direction of action.

Applying Newton's Laws

Equilibrium of a Particle

• For equilibrium: $\sum \vec{F} = 0$

• Resolve forces into components; set $\sum F_x = 0$, $\sum F_y = 0$

• Use free-body diagrams to identify all forces.

Dynamics of Particles

• For non-equilibrium (acceleration): $\sum \vec{F} = m \vec{a}$

• Choose axes so that one is along the direction of acceleration.

• Apply Newton's second law to each object separately if more than one object is involved.

Friction Forces

• Kinetic Friction: $f_k = \mu_k n$ (opposes sliding motion)

• Static Friction: $f_s \leq \mu_s n$ (prevents motion up to a maximum value)

• Friction and normal forces are always perpendicular.

• Rolling friction and fluid resistance are also important in real systems.

Table: Typical Coefficients of Friction

Surface Pair	$\mu_s$	$\mu_k$
Rubber on dry concrete	1.0	0.8
Steel on steel	0.74	0.57
Teflon on Teflon	0.04	0.04

Fluid Resistance and Terminal Speed

• At low speeds: $f = kv$

• At high speeds: $f = Dv^2$

• Terminal speed (for $f = kv$): $v_t = \frac{mg}{k}$

• Terminal speed (for $f = Dv^2$): $v_t = \sqrt{\frac{mg}{D}}$

Dynamics of Circular Motion

• For uniform circular motion: $F_{\text{net}} = m a_{\text{rad}} = m \frac{v^2}{R}$

• On a flat curve, friction provides the centripetal force: $v_{\text{max}} = \sqrt{\mu_s g R}$

• On a banked curve (no friction): $\tan \theta = \frac{v^2}{gR}$

• For vertical circles, consider weight and normal/tension forces at top and bottom.

The Fundamental Forces of Nature

• Gravitational: Attraction between masses; weakest but long-range.

• Electromagnetic: Forces between charges; includes contact forces, friction, tension, normal force.

• Strong Nuclear: Holds atomic nuclei together; very short range, very strong.

• Weak Nuclear: Responsible for certain types of radioactive decay; very short range.

Summary Table: Four Fundamental Forces

Force	Relative Strength	Range	Role
Gravitational	Weakest	Infinite	Planets, stars, galaxies
Electromagnetic	~1036 times gravity	Infinite	Atoms, molecules, chemistry
Strong Nuclear	Strongest	~10-15 m	Holds nuclei together
Weak Nuclear	~10-6 of strong	~10-17 m	Radioactive decay

Key Equations

• $v = v_0 + a t$

• $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• $v^2 = v_0^2 + 2 a (x - x_0)$

• $\vec{A} = A_x \hat{\imath} + A_y \hat{\jmath} + A_z \hat{k}$

• $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$

• $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y) \hat{\imath} + (A_z B_x - A_x B_z) \hat{\jmath} + (A_x B_y - A_y B_x) \hat{k}$

• $f_k = \mu_k n$

• $f_s \leq \mu_s n$

• $F_{\text{net}} = m a$

• $w = m g$

• $F_{\text{gravity}} = G \frac{m_1 m_2}{r^2}$ (see Ch. 13)

Additional info:

• These notes cover the foundational concepts and problem-solving strategies for the first five chapters of a calculus-based college physics course, including units, vectors, kinematics, Newton's laws, friction, and circular motion.

• For more advanced topics (energy, momentum, rotation, etc.), see subsequent chapters.