Physics 2025 Exam 1 Formula Sheet: Essential Equations and Concepts

Fundamental Constants and Geometric Formulas

Basic Constants and Geometry in Physics

Physics problems often require the use of fundamental constants and geometric formulas. These are essential for calculations involving circles, spheres, and rotational motion.

• Pi ($\pi$): $\pi = 3.14$ (approximate value)

• Circumference of a Circle: $C = 2\pi r$

• Area of a Circle: $A_{circle} = \pi r^2$

• Surface Area of a Sphere: $A_{sphere} = 4\pi r^2$

• Revolution and Degrees: $1\,\text{rev} = 2\pi = 360^\circ$

Physical Constants

Gravitational Acceleration and Universal Gravitational Constant

These constants are used in calculations involving gravity and motion near Earth's surface or in universal gravitation problems.

• Acceleration due to Gravity: $g = 9.8\,\text{m}/\text{s}^2$

• Universal Gravitational Constant: $G = 6.67 \times 10^{-11}\,\text{N}\,\text{m}^2/\text{kg}^2$

Kinematics

Time, Velocity, and Acceleration

Kinematics describes the motion of objects using displacement, velocity, and acceleration. The following equations are fundamental for analyzing linear motion.

• Elapsed Time: $\Delta t = t - t_i$

• Average Velocity: $v_{av,x} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$

• Average Acceleration: $a_{av,x} = \frac{\Delta v_x}{\Delta t} = \frac{v_{f,x} - v_{i,x}}{t_f - t_i}$

Kinematic Equations for Constant Acceleration

• Final Velocity: $v_x = v_{i,x} + a_x \Delta t$

• Displacement: $x = x_i + v_{i,x} \Delta t + \frac{1}{2} a_x (\Delta t)^2$

• Velocity-Squared Equation: $v_x^2 = v_{i,x}^2 + 2a_x(x - x_0)$

• Average Velocity (for constant acceleration): $v_{av,x} = \frac{v_{i,x} + v_{f,x}}{2}$

Projectile Range Equation

• Horizontal Range: $x = \frac{v_i^2 \sin(2\theta)}{g}$

• Application: Used to calculate the distance a projectile travels horizontally when launched at an angle $\theta$ with initial speed $v_i$.

Quadratic Formula

Solving Quadratic Equations

Quadratic equations frequently arise in physics, especially in kinematics and energy problems.

• General Form: $0 = ax^2 + bx + c$

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

• Example: Used to solve for time or position when displacement equations are quadratic in form.

Trigonometry in Physics

Right Triangle Relationships

Trigonometric functions are essential for resolving vectors and analyzing motion in two dimensions.

• Sine: $\sin \theta = \frac{\text{opp}}{\text{hyp}}$

• Cosine: $\cos \theta = \frac{\text{adj}}{\text{hyp}}$

• Tangent: $\tan \theta = \frac{\text{opp}}{\text{adj}}$

• Pythagorean Theorem: $a^2 + b^2 = c^2$

• Application: Used to resolve vector components and calculate distances.

Forces and Newton's Laws

Net Force and Newton's Second Law

Newton's laws describe the relationship between forces and motion. The following equations are used to analyze forces in different directions.

• Net Force (x-direction): $F_{net,x} = m a_x$

• Net Force (y-direction): $F_{net,y} = m a_y$

Gravitational Force

• Weight: $F_g = m g$

• Universal Law of Gravitation: $F_g = G \frac{m_1 m_2}{r^2}$

Frictional Forces

• Kinetic Friction: $f_k = \mu_k F_N$

• Maximum Static Friction: $f_{s,max} = \mu_s F_N$

• Application: Used to determine the force required to start or maintain motion on a surface.

Circular Motion

Centripetal Acceleration and Velocity

Objects moving in circles experience centripetal acceleration and velocity, which are described by the following equations.

• Centripetal Acceleration: $a_R = \frac{v^2}{r}$

• Velocity in Circular Motion: $v = \frac{2\pi r}{T}$

• Period and Frequency: $T = \frac{1}{f}$

• Application: Used in problems involving rotation, orbits, and periodic motion.

Summary Table: Key Equations and Their Applications

Equation	Application
$C = 2\pi r$	Circumference of a circle
$A_{circle} = \pi r^2$	Area of a circle
$A_{sphere} = 4\pi r^2$	Surface area of a sphere
$g = 9.8\,\text{m}/\text{s}^2$	Acceleration due to gravity
$G = 6.67 \times 10^{-11}\,\text{N}\,\text{m}^2/\text{kg}^2$	Universal gravitational constant
$F_g = m g$	Weight of an object
$F_g = G \frac{m_1 m_2}{r^2}$	Gravitational force between two masses
$f_k = \mu_k F_N$	Kinetic friction force
$f_{s,max} = \mu_s F_N$	Maximum static friction
$a_R = \frac{v^2}{r}$	Centripetal acceleration
$v = \frac{2\pi r}{T}$	Velocity in circular motion
$T = \frac{1}{f}$	Period and frequency relationship
$x = \frac{v_i^2 \sin(2\theta)}{g}$	Projectile range
$x = x_i + v_{i,x} \Delta t + \frac{1}{2} a_x (\Delta t)^2$	Displacement with constant acceleration
$v_x^2 = v_{i,x}^2 + 2a_x(x - x_0)$	Velocity-squared equation
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	Quadratic formula

Additional info: This formula sheet covers foundational equations for introductory college physics, including kinematics, forces, circular motion, and basic trigonometry. These equations are essential for solving problems in classical mechanics and are commonly provided for reference during exams.