Exhaustive Comprehensive Physics University Study Guide: University Physics and AP Physics Principles

Scalar Definition: A quantity described by magnitude only.

Examples: Mass, time, distance, speed, temperature, volume, charge.

Vector Definition: A quantity described by both magnitude and direction.

Examples: Displacement, velocity, acceleration, force, momentum, position.

Vector Characteristics:

• Direction is critical: $3\,km$ east is not equal to $3\,km$ west ($3\,km\, ext{east} \neq 3\,km\, ext{west}$).

• Negative signs on vectors indicate a direction opposite to the defined positive direction.

Vector Components:

• Vectors can be decomposed into orthogonal components:

• Horizontal component: $x = v\cos(\theta)$

• Vertical component: $y = v\sin(\theta)$

• Unit vector notation: $\mathbf{A} = A<em>x\hat{i} + A</em>y\hat{j}$

• Vector magnitude determination: $|\mathbf{A}| = \sqrt{A<em>x^2 + A</em>y^2}$

• Vector direction (angle) determination: $\theta = \tan^{-1}\left(\frac{A<em>y}{A</em>x}\right)$

Kinematics and Motion Analysis:

Position ($\mathbf{x}$): The location of an object relative to a specific reference point in a coordinate system.

Distance vs. Displacement:

• Distance: The total path length traveled (scalar).

• Displacement ($\Delta x$): The change in position, as calculated by $\Delta x = x<em>f - x</em>i$ (vector).

Speed vs. Velocity:

• Speed: Distance divided by time (scalar).

• Velocity: Displacement divided by time (vector).

• Average Velocity: $\mathbf{v}_{avg} = \frac{\Delta x}{\Delta t}$

Instantaneous Velocity: The velocity at a specific instant in time; represented graphically as the slope of the position-time graph.

Acceleration ($\mathbf{a}$): The change in velocity over time, including changes in either speed or direction.

Unit: $m/s^2$

Average Acceleration: $\mathbf{a} = \frac{\Delta \mathbf{v}}{\Delta t}$

Directional Relationships: If $\mathbf{v}$ and $\mathbf{a}$ direct the same, the object speeds up; if opposing, the object slows down.

Types of Motion:

• Uniform Motion: Constant velocity with a straight-line position-time graph where $x \propto t$.

• Linear (Translational): Movement through space along a straight line.

• Circular (Translational): Moving along a circular path.

• Projectile Motion: 2D motion influenced by gravity alone.

• Rotational Motion: Spinning about a fixed internal axis.

Trajectory: The specific path followed by a moving object.

Kinematics Equations (Constant Acceleration):

1. $v = v_0 + at$

2. $x = \frac{(v + v_0)}{2}t$

3. $x = v_0t + \frac{1}{2}at^2$

4. $v^2 = v_0^2 + 2ax$

Free Fall:

• Motion where gravity is the sole acting force (ignoring air resistance).

• Magnitude of free-fall acceleration: $g = 9.8\,m/s^2$.

• Vertical acceleration: $a_y = -g$.

• At the peak of vertical motion, instantaneous vertical velocity $v_y = 0$.

Projectile Motion (2D):

• Horizontal and vertical components are independent.

• Horizontal motion: Constant velocity ($v_x = v\cos(\theta)$).

• Vertical motion: Constant acceleration ($v_y = v\sin(\theta)$).