Vectors and Two-Dimensional Kinematics

Vectors and 2-Dimensional Kinematics

Definition of Vectors

Vectors are fundamental quantities in physics that possess both magnitude and direction. They are typically represented by arrows, where the length of the arrow indicates the magnitude and the direction of the arrow shows the direction of the vector.

• Notation: Vectors are often denoted with an arrow above the letter (e.g., \( \vec{A} \)), and their magnitude is written as \( A = |\vec{A}| \).

• Example: Displacement, velocity, and acceleration are all vector quantities.

Unit Vectors and Unit Vector Notation

Unit vectors are vectors of length one that indicate direction along coordinate axes. In two dimensions, the standard unit vectors are:

• \( \hat{i} \): Unit vector in the x-direction

• \( \hat{j} \): Unit vector in the y-direction

Any vector in the plane can be written as:

• \( \vec{A} = A_x \hat{i} + A_y \hat{j} \)

Vector Components

Vectors can be resolved into components along the x and y axes. If a vector \( \vec{A} \) makes an angle \( \theta \) with the x-axis:

• \( A_x = A \cos \theta \)

• \( A_y = A \sin \theta \)

• \( \vec{A} = A_x \hat{i} + A_y \hat{j} \)

For vectors in other quadrants, the signs of the components must be considered based on direction.

Magnitude and Direction of a Vector

The magnitude and direction of a vector given its components are:

• \( A = \sqrt{A_x^2 + A_y^2} \)

• \( \tan \theta = \left| \frac{A_y}{A_x} \right| \)

Example: Displacement Vector in Unit Vector Notation

Given a displacement of 5 km directed \( \theta = 53^\circ \) East of South:

• \( D_x = +D \sin \theta = 5 \text{ km} \times 0.8 = 4 \text{ km} \)

• \( D_y = -D \cos \theta = -5 \text{ km} \times 0.6 = -3 \text{ km} \)

• \( \vec{D} = 4 \hat{i} + (-3) \hat{j} \) km

Vector Addition and Subtraction

Graphical Method

• Vectors are added by placing them head-to-tail and drawing the resultant from the tail of the first to the head of the last.

• Subtraction is performed by adding the negative of a vector.

Component Method

• \( \vec{A} = A_x \hat{i} + A_y \hat{j} \)

• \( \vec{B} = B_x \hat{i} + B_y \hat{j} \)

• \( \vec{C} = \vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} \)

• \( \vec{D} = \vec{A} - \vec{B} = (A_x - B_x)\hat{i} + (A_y - B_y)\hat{j} \)

Position, Velocity, and Acceleration in 2-D

Position Vector

• \( \vec{r} = x \hat{i} + y \hat{j} \)

Velocity

• Average velocity: \( \vec{v}_{av} = \frac{\Delta \vec{r}}{\Delta t} \)

• Instantaneous velocity: \( \vec{v} = \frac{d\vec{r}}{dt} = v_x \hat{i} + v_y \hat{j} \)

• \( v_x = \frac{dx}{dt} \), \( v_y = \frac{dy}{dt} \)

Acceleration

• \( \vec{a} = \frac{d\vec{v}}{dt} = a_x \hat{i} + a_y \hat{j} \)

• \( a_x = \frac{dv_x}{dt} = \frac{d^2x}{dt^2} \)

• \( a_y = \frac{dv_y}{dt} = \frac{d^2y}{dt^2} \)

Effect of Acceleration Components

• Acceleration parallel to velocity (\( a_{||} \)) changes the magnitude (speed).

• Acceleration perpendicular to velocity (\( a_{\perp} \)) changes the direction.

Projectile Motion

Projectile motion is a classic example of two-dimensional kinematics where the only acceleration is due to gravity (free fall):

• \( a_x = 0 \)

• \( a_y = -g \)

• Velocity components:

  • \( v_x = v_{0x} + a_x t = v_{0x} \)

  • \( v_y = v_{0y} + a_y t = v_{0y} - gt \)

Demonstrations and Applications

• Vertical launch of a ball from a moving car demonstrates independence of x and y motion.

• Simultaneous dropping and horizontal launching of balls shows that both reach the ground at the same time if released from the same height.

Additional info: These notes cover the foundational aspects of vectors and two-dimensional kinematics, which are essential for understanding more advanced topics in mechanics and physics in general.