Kinematics and Motion: Essential Formulas and Concepts for Physics with Calculus

Kinematics and Motion

Coordinate Systems and Units

Choosing and maintaining a consistent coordinate system is fundamental in physics problems. Units must always be checked for consistency, and significant digits should be managed carefully for accuracy.

• Coordinate System: Select a reference direction (e.g., positive x-axis) and stick with it throughout the problem.

• Unit Conversions: Multiply by ratios of equal quantities to cancel original units and convert to desired units.

• Significant Digits: Keep 4 digits in intermediate calculations; round final answers to 3 or 2 digits.

Definitions of Velocity and Acceleration

Motion in one dimension is described using displacement, velocity, and acceleration. These quantities are defined as follows:

• Average Velocity: The rate of change of displacement over time. $v_{\text{av}} = \frac{\Delta x}{\Delta t}$

• Instantaneous Velocity: The slope of the position vs. time graph at a specific point.

• Average Acceleration: The rate of change of velocity over time. $a_{\text{av}} = \frac{\Delta v}{\Delta t}$ where $\Delta v = v_f - v_i$

• Vectors in 2D/3D: Position, displacement, velocity, and acceleration are all vector quantities in higher dimensions.

Kinematics Equations

Kinematics describes motion using equations that relate displacement, velocity, acceleration, and time. These equations are valid for constant velocity or constant acceleration.

• Constant Velocity: $\Delta x = v \Delta t$ $x(t) = x_0 + v t$

• Constant Acceleration: $\Delta v = a \Delta t$ $v(t) = v_0 + a t$ $v_{\text{av}} = \frac{1}{2}(v_i + v_f)$ $\Delta y = v_0 t + \frac{1}{2} a t^2$ $y(t) = y_0 + v_0 t + \frac{1}{2} a t^2$ $v_f^2 = v_i^2 + 2 a \Delta y$

Projectile Motion

Projectile motion involves both horizontal and vertical components. The horizontal motion is at constant velocity, while the vertical motion is under constant acceleration due to gravity.

• Horizontal Motion: $x$ direction, constant velocity.

• Vertical Motion: $y$ direction, constant acceleration $-g$ (downward).

• Gravity: $g = 9.8\ \text{m/s}^2$

• Initial Velocity Components: Use trigonometry (SOHCAHTOA) to find $x$ and $y$ components. Measure angle counterclockwise from the $+x$ axis.

• Angle and Components: $\tan \theta = \frac{y}{x}$ (be careful if $x$ is negative).

Vector Addition

Vectors can be added graphically or algebraically. In algebraic addition, add corresponding components.

• Graphical Addition: Tip-to-tail method.

• Algebraic Addition: Add $x$ and $y$ components separately.

Relative Motion

Relative motion describes how the velocity of one object appears from the perspective of another.

• Relative Velocity Formula: $\vec{v}_{A\ \text{rel}\ C} = \vec{v}_{A\ \text{rel}\ B} + \vec{v}_{B\ \text{rel}\ C}$ or $\vec{v}_{A\ \text{rel}\ B} - \vec{v}_{C\ \text{rel}\ B}$

Quadratic Formula

The quadratic formula is used to solve equations of the form $ax^2 + bx + c = 0$.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Unit Conversions

Unit conversions are essential for ensuring consistency in calculations. Common length conversions are provided below.

Unit	Equivalent
1 meter	1.0936 yard
1 meter	3.2808 feet
1 meter	39.37 inch
1 km	1000 m
1 mile	1609 m

Examples and Applications

• Example 1: Calculating average velocity for a car traveling 100 m in 5 s: $v_{\text{av}} = \frac{100\ \text{m}}{5\ \text{s}} = 20\ \text{m/s}$

• Example 2: Projectile launched at 30° with initial speed 20 m/s: Find $v_x = v_0 \cos \theta$, $v_y = v_0 \sin \theta$. $v_x = 20 \cos 30^\circ = 17.32\ \text{m/s}$ $v_y = 20 \sin 30^\circ = 10\ \text{m/s}$

Additional info: Academic context was added to clarify vector addition, projectile motion, and the use of the quadratic formula in kinematics problems.