Chapter 1: Units, Physical Quantities, and Vectors

Metric Units, Prefixes, and Conversions

The metric system is the standard system of measurement in physics, using base units such as meters (m), kilograms (kg), and seconds (s). Prefixes denote powers of ten to simplify large or small quantities.

• Base Units: Meter (m) for length, kilogram (kg) for mass, second (s) for time.

• Common Prefixes: kilo- (k, ), centi- (c, ), milli- (m, ), micro- (, ), nano- (n, ).

• Unit Conversion: Multiply by the appropriate power of ten to convert between units.

• Example:

Significant Figures and Scientific Notation

Significant figures reflect the precision of a measurement. Scientific notation expresses numbers as a product of a coefficient and a power of ten.

• Significant Figures: All nonzero digits are significant; zeros between nonzero digits are significant; leading zeros are not significant.

• Scientific Notation: , where and is an integer.

• Example: (three significant figures)

Vector Arithmetic: Addition, Subtraction, Multiplication, Division

Vectors have both magnitude and direction. Operations include addition, subtraction, and multiplication by scalars.

• Addition/Subtraction: Combine components:

• Multiplication by Scalar:

• Division by Scalar:

• Example: If ,

Vector Components and Unit Vectors

Any vector can be expressed in terms of its components along the coordinate axes using unit vectors , , and .

• Component Form:

• Magnitude:

• Direction:

• Example: has magnitude

Scalar (Dot) Product and Vector (Cross) Product

Two main ways to multiply vectors: the dot product (scalar) and the cross product (vector).

• Dot Product: (results in a scalar)

• Cross Product: (results in a vector perpendicular to both)

• Example: ,

Chapter 2: Motion Along a Straight Line

Distance and Displacement

Distance is the total length traveled, while displacement is the straight-line change in position.

• Displacement:

• Distance: Always positive; does not depend on direction.

• Example: Walking 3 m east, then 4 m west: distance = 7 m, displacement = -1 m

Speed and Velocity

Speed is the rate of distance traveled; velocity is the rate of displacement (a vector).

• Average Speed:

• Average Velocity:

• Instantaneous Velocity:

Acceleration

Acceleration is the rate of change of velocity.

• Average Acceleration:

• Instantaneous Acceleration:

• Example: If velocity increases from 0 to 10 m/s in 2 s,

Average vs. Instantaneous Quantities

Average quantities are measured over a finite interval; instantaneous quantities are measured at a specific moment.

• Average Velocity:

• Instantaneous Velocity:

Graphs: x vs. t, v vs. t

Graphs are useful for visualizing motion.

• x vs. t: Slope gives velocity.

• v vs. t: Slope gives acceleration; area under curve gives displacement.

• Example: A straight line in x vs. t means constant velocity.

Equations for Constant Acceleration

For motion with constant acceleration, the following kinematic equations apply:

• Example: Dropping a ball from rest (),

Free Fall

Objects in free fall experience constant acceleration due to gravity ( downward).

• Equations: Same as constant acceleration, with (downward).

• Example: Time to fall from height :

Integration: Velocity from Acceleration, Position from Velocity

Integration allows calculation of velocity and position from acceleration when acceleration is not constant.

• Example: If , then

Chapter 3: Motion in Two or Three Dimensions

Vector Expressions of Position, Velocity, and Acceleration

In multiple dimensions, position, velocity, and acceleration are expressed as vectors with components.

• Position:

• Velocity:

• Acceleration:

Components of Acceleration: Tangential (Parallel) vs. Radial (Perpendicular)

Acceleration can be decomposed into tangential (along the path) and radial (toward the center of curvature) components.

• Tangential Acceleration (): Changes the speed along the path.

• Radial (Centripetal) Acceleration (): Changes the direction of velocity; points toward the center of curvature.

• Formulas: ,

Equations of Projectile Motion

Projectile motion involves two-dimensional motion under constant acceleration (gravity).

• Horizontal Motion:

• Vertical Motion:

• Range:

• Example: A ball launched at angle with speed follows a parabolic path.

Uniform and Non-Uniform Circular Motion: Centripetal and Tangential Acceleration

In circular motion, the acceleration can have radial (centripetal) and tangential components.

• Uniform Circular Motion: Speed is constant; only centripetal acceleration present.

• Non-Uniform Circular Motion: Speed changes; both centripetal and tangential acceleration present.

• Centripetal Acceleration:

• Tangential Acceleration:

• Example: A car turning in a circle at increasing speed experiences both and .