Equation Sheet for Introductory Physics: Kinematics, Dynamics, and Trigonometry

Trigonometric Identities and Algebraic Equations

Basic Trigonometric Ratios

Trigonometric functions are essential for resolving vectors and analyzing motion in physics. The following are the fundamental definitions:

• Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Key Trigonometric Identities

• $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$

• $\sin 2\theta = 2 \sin \theta \cos \theta$

• $\cos^2 \theta + \sin^2 \theta = 1$

Quadratic Formula

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

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

Basic Integration Formulas

• $\int a^n da = \frac{a^{n+1}}{n+1} + c$

• $\int_{a_1}^{a_2} a^n da = \frac{a_2^{n+1}}{n+1} - \frac{a_1^{n+1}}{n+1}$

Gravitational Acceleration

• $g = 9.8\ \mathrm{m/s^2}$ (standard value near Earth's surface)

Kinematics in One and Two Dimensions

Position, Velocity, and Acceleration

Kinematics describes the motion of objects without reference to the forces causing the motion. The following equations relate position, velocity, and acceleration:

• Velocity as the derivative of position: $v_x = \frac{dx}{dt}$

• Displacement from velocity: $\int_{x_1}^{x_2} dx = \int_{t_1}^{t_2} v_x dt$

• Acceleration as the derivative of velocity: $a_x = \frac{dv_x}{dt}$

• Change in velocity from acceleration: $\int_{v_{x1}}^{v_{x2}} dv_x = \int_{t_1}^{t_2} a_x dt$

Kinematic Equations for Constant Acceleration

For motion with constant acceleration, the following equations are used (subscripts $i$ and $f$ denote initial and final values):

• $v_{xf} = v_{xi} + a_x \Delta t$

• $\Delta x = v_{xi} \Delta t + \frac{1}{2} a_x (\Delta t)^2$

• $\Delta x = v_{xf} \Delta t - \frac{1}{2} a_x (\Delta t)^2$

• $v_{xf}^2 = v_{xi}^2 + 2 a_x \Delta x$

• $\Delta x = \frac{v_{xf} + v_{xi}}{2} \Delta t$

For vertical motion (y-direction), replace $x$ with $y$ and $a_x$ with $a_y$ as appropriate.

Example:

A ball is thrown straight up with an initial velocity $v_{yi} = 10\ \mathrm{m/s}$. How high does it go?

• Use $v_{yf}^2 = v_{yi}^2 + 2 a_y \Delta y$ with $v_{yf} = 0$ at the top, $a_y = -9.8\ \mathrm{m/s^2}$.

• $0 = (10)^2 + 2(-9.8)\Delta y \implies \Delta y = \frac{-100}{2 \times -9.8} \approx 5.1\ \mathrm{m}$

Newton's Laws of Motion and Forces

Newton's Second Law

Newton's Second Law relates the net force acting on an object to its acceleration:

• $\sum F_x = m a_x$

• $\sum F_y = m a_y$

• In vector form: $\sum \vec{F} = m \vec{a}$

Friction Forces

• Static friction: $0 \leq f_s \leq \mu_s n$

• Kinetic friction: $f_k = \mu_k n$

• Normal force: $F_N = mg$ (for horizontal surfaces)

Hooke's Law (Spring Force)

• $\vec{F}_s = -k \Delta \vec{x}$

• Where $k$ is the spring constant and $\Delta \vec{x}$ is the displacement from equilibrium.

Example:

A 2 kg block rests on a horizontal surface ($\mu_s = 0.5$). What is the maximum static friction force?

• $F_N = mg = 2 \times 9.8 = 19.6\ \mathrm{N}$

• $f_{s,\text{max}} = \mu_s F_N = 0.5 \times 19.6 = 9.8\ \mathrm{N}$

Summary Table: Key Equations

Concept	Equation	Description
Trigonometric Ratios	$\sin \theta, \cos \theta, \tan \theta$	Relate sides of a right triangle
Kinematic Equations	$v_{xf} = v_{xi} + a_x \Delta t$ $\Delta x = v_{xi} \Delta t + \frac{1}{2} a_x (\Delta t)^2$ $v_{xf}^2 = v_{xi}^2 + 2 a_x \Delta x$	Constant acceleration motion
Newton's Second Law	$\sum \vec{F} = m \vec{a}$	Net force causes acceleration
Friction	$f_s \leq \mu_s n$ $f_k = \mu_k n$	Static and kinetic friction
Hooke's Law	$\vec{F}_s = -k \Delta \vec{x}$	Force by a spring

Additional info: Some equations and context were inferred and expanded for clarity and completeness, including the use of standard notation and example applications.