Physics I: 2D Kinematics and Relative Velocity – Study Notes

2D Kinematics

Introduction to 2D Kinematics

Two-dimensional (2D) kinematics studies the motion of objects in a plane, considering both the x (horizontal) and y (vertical) directions. This topic is fundamental in understanding projectile motion and the decomposition of vectors.

• Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion (forces).

• In 2D kinematics, position, velocity, and acceleration are treated as vectors with both magnitude and direction.

Vector Representation and Components

Vectors in 2D can be broken down into x and y components, which are analyzed separately.

• Displacement vector $\vec{r}$: Represents the change in position.

• Velocity vector $\vec{v}$: The rate of change of displacement.

• For a vector $\vec{v}$ at angle $\theta$ from the x-axis:

  • $v_x = v \cos \theta$

  • $v_y = v \sin \theta$

• Graphical representation: Vectors are often drawn as arrows on coordinate axes, with their components shown as projections onto the axes.

Equations of Motion in 2D

The motion in each direction (x and y) can be described using the kinematic equations, assuming constant acceleration (such as gravity in projectile motion).

• Horizontal motion (x-direction):

  • Acceleration $a_x = 0$ (if no air resistance)

  • Velocity: $v_x = v_{0x}$ (constant)

  • Displacement: $x = x_0 + v_{0x} t$

• Vertical motion (y-direction):

  • Acceleration $a_y = -g$ (where $g$ is the acceleration due to gravity, $9.8\,\mathrm{m/s^2}$ downward)

  • Velocity: $v_y = v_{0y} - g t$

  • Displacement: $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

Projectile Motion

Projectile motion is a common example of 2D kinematics, where an object is launched into the air and moves under the influence of gravity alone.

• Initial velocity components:

  • $v_{0x} = v_0 \cos \theta$

  • $v_{0y} = v_0 \sin \theta$

• Time of flight (for projectiles launched and landing at the same height):

  • $t_f = \frac{2 v_{0y}}{g}$

• Maximum height:

  • $h_{max} = \frac{v_{0y}^2}{2g}$

• Range (horizontal distance traveled):

  • $R = v_{0x} t_f = \frac{v_0^2 \sin 2\theta}{g}$

• Velocity at any time:

  • $v_x = v_{0x}$

  • $v_y = v_{0y} - g t$

  • Total speed: $v = \sqrt{v_x^2 + v_y^2}$

  • Direction: $\alpha = \tan^{-1}\left(\frac{v_y}{v_x}\right)$

Example: A ball is launched at $v_0 = 10\,\mathrm{m/s}$ at an angle $\theta = \frac{\pi}{3}$ above the horizontal. Find the time of flight, maximum height, and range.

• Calculate $v_{0x}$ and $v_{0y}$ using the angle.

• Apply the above formulas for $t_f$, $h_{max}$, and $R$.

Summary Table: Key Equations for Projectile Motion

Quantity	Equation
Horizontal displacement	$x = v_{0x} t$
Vertical displacement	$y = v_{0y} t - \frac{1}{2} g t^2$
Time of flight	$t_f = \frac{2 v_{0y}}{g}$
Maximum height	$h_{max} = \frac{v_{0y}^2}{2g}$
Range	$R = \frac{v_0^2 \sin 2\theta}{g}$
Velocity components	$v_x = v_{0x}$, $v_y = v_{0y} - g t$
Speed at time $t$	$v = \sqrt{v_x^2 + v_y^2}$

Relative Velocity

Introduction to Relative Velocity

Relative velocity describes the velocity of one object as observed from another moving object. This concept is essential when analyzing motion in different reference frames, such as a boat moving in a river or an airplane in the wind.

• Relative velocity is a vector difference between the velocities of two objects as measured in a common reference frame.

• Notation: $\vec{v}_{AB}$ is the velocity of A relative to B.

Relative Velocity Equations

• General formula: $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$

• For three objects (e.g., person, boat, and water):

  • $\vec{v}_{PD} = \vec{v}_{PB} + \vec{v}_{BD}$

  • Where $\vec{v}_{PD}$ is the velocity of the person relative to the dock, $\vec{v}_{PB}$ is the velocity of the person relative to the boat, and $\vec{v}_{BD}$ is the velocity of the boat relative to the dock.

• Vector addition is used to combine velocities in different directions.

Example: Boat in a River

• A boat moves east at $5\,\mathrm{m/s}$ relative to the water, and the water flows north at $3\,\mathrm{m/s}$ relative to the ground. The velocity of the boat relative to the ground is found by vector addition:

  • $\vec{v}_{BG} = \vec{v}_{BW} + \vec{v}_{WG}$

  • Magnitude: $v = \sqrt{5^2 + 3^2} = 5.83\,\mathrm{m/s}$

  • Direction: $\tan \alpha = \frac{3}{5}$, $\alpha = 31^\circ$ north of east

Summary Table: Relative Velocity Notation

Symbol	Meaning
$\vec{v}_{AB}$	Velocity of A relative to B
$\vec{v}_{BW}$	Velocity of boat relative to water
$\vec{v}_{WS}$	Velocity of water relative to shore
$\vec{v}_{BS}$	Velocity of boat relative to shore ($\vec{v}_{BW} + \vec{v}_{WS}$)

Additional info: Some equations and diagrams were inferred from standard 2D kinematics and relative velocity problems, as the handwritten notes were partially fragmented.