Physics Equation Sheet: Kinematics, Vectors, and Projectile Motion

Kinematics and Vectors

Fundamental Constants and Trigonometric Values

This section provides essential constants and trigonometric identities frequently used in introductory physics, especially in kinematics and vector analysis.

• Acceleration due to gravity (g): $g = 9.8\ \text{m/s}^2$

• Trigonometric values:

  • $\sin(30^\circ) = \cos(60^\circ) = \frac{1}{2}$

  • $\sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}$

  • $\cos(30^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2}$

Vector Components and Operations

Vectors are quantities with both magnitude and direction. They can be decomposed into components and combined using vector addition.

• Magnitude of a vector: $A = \sqrt{A_x^2 + A_y^2}$

• Components of a vector:

  • $A_x = A \cos \theta$

  • $A_y = A \sin \theta$

• Direction (angle) of a vector: $\tan \theta = \frac{A_y}{A_x}$

• Vector addition: $\vec{C} = \vec{A} + \vec{B}$

• Component addition:

  • $C_x = A_x + B_x$

  • $C_y = A_y + B_y$

Quadratic Equation

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

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

Kinematic Equations

Average Velocity and Acceleration

Kinematics describes the motion of objects using position, velocity, and acceleration.

• Average velocity: $\vec{v}_{avg} = \frac{\Delta \vec{x}}{\Delta t}$

• Average acceleration: $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$

Equations of Motion (Constant Acceleration)

These equations relate displacement, velocity, acceleration, and time for objects moving with constant acceleration.

• $v_{fx} = v_{ix} + a_x \Delta t$

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

• $v_{fx}^2 = v_{ix}^2 + 2 a_x \Delta x$

Vector Addition in Motion

• $\vec{v}_{CB} = \vec{v}_{CA} + \vec{v}_{AB}$

Projectile Motion

Trajectory and Range Equations

Projectile motion involves two-dimensional motion under gravity, with horizontal and vertical components analyzed separately.

• Equation for trajectory: $y(x) = (\tan \theta_i)x - \frac{1}{2} \frac{g x^2}{(v_i \cos \theta_i)^2}$

• Time of flight: $t = \frac{v_i \sin \theta_i}{g}$

• Maximum height: $h = \frac{v_i^2 \sin^2 \theta_i}{2g}$

• Range: $R = \frac{v_i^2 \sin 2\theta_i}{g}$

Circular Motion

Frequency, Velocity, and Centripetal Acceleration

Circular motion involves objects moving in a circle at constant speed, characterized by frequency, velocity, and centripetal acceleration.

• Frequency: $f = \frac{1}{T}$

• Velocity: $v = \frac{2\pi r}{T} = 2\pi r f$

• Centripetal acceleration: $a_c = \frac{v^2}{r}$

Summary Table: Key Equations and Their Applications

Equation	Application
$A = \sqrt{A_x^2 + A_y^2}$	Magnitude of a vector
$\tan \theta = \frac{A_y}{A_x}$	Direction of a vector
$v_{fx} = v_{ix} + a_x \Delta t$	Final velocity (constant acceleration)
$\Delta x = v_{ix} \Delta t + \frac{1}{2} a_x (\Delta t)^2$	Displacement (constant acceleration)
$v_{fx}^2 = v_{ix}^2 + 2 a_x \Delta x$	Velocity-displacement relation
$y(x) = (\tan \theta_i)x - \frac{1}{2} \frac{g x^2}{(v_i \cos \theta_i)^2}$	Projectile trajectory
$R = \frac{v_i^2 \sin 2\theta_i}{g}$	Projectile range
$a_c = \frac{v^2}{r}$	Centripetal acceleration

Additional info:

• All equations assume ideal conditions (e.g., no air resistance for projectile motion).

• Vector notation ($\vec{A}$) indicates directionality; scalar quantities lack direction.

• Trigonometric identities are essential for resolving vectors into components.