Introduction to Two-Dimensional Vector Kinematics

Conversely, if you have the Cartesian coordinates $(x, y)$, you can find the magnitude $r$ using the Pythagorean theorem and the angle $\theta$ using the tangent function:

$$\begin{aligned} r &= \sqrt{x^2 + y^2} \\ \theta &= \tan^{-1}\left(\frac{y}{x}\right) \end{aligned}$$

Exam Tip: When calculating $\theta$ using $\tan^{-1}(y/x)$, always check the signs of your $x$ and $y$ components to determine which quadrant the vector is in. The calculator might give you an angle in the wrong quadrant if you don't manually adjust for the signs of $x$ and $y$.

Vector Notation and Components

Once we have a coordinate system, we need a way to write down our vectors. There are two common ways to express them: magnitude and direction notation, and unit vector notation.

Magnitude and Direction Notation

This is the most "natural" way to describe a vector in everyday language. For example, "a force of 10 N at an angle of $30^\circ$." It is often written as $(10\text{ N}, 30^\circ)$.

Unit Vector Notation

In formal physics calculations, we use unit vector notation. This involves using the unit vectors $\hat{i}$ (pointing in the $+x$ direction) and $\hat{j}$ (pointing in the $+y$ direction). A vector $\vec{A}$ is written as:

$$\vec{A} = A_x \hat{i} + A_y \hat{j}$$

Where $A_x$ and $A_y$ are the components of the vector along the respective axes. Unit vector notation is incredibly powerful because it simplifies vector addition, subtraction, and multiplication.

Separation of Motion into Components

One of the most critical concepts in kinematics is the separation of motion into components. This principle states that the motion in the $x$-direction is independent of the motion in the $y$-direction.

For example, if a skydiver is falling and moving horizontally at the same time, you can calculate her vertical displacement using only her vertical velocity, without needing to worry about her horizontal speed. This allows us to take complex two-dimensional problems and turn them into two simple one-dimensional problems.

Analyzing Position and Velocity

In two dimensions, we track an object's movement using the position vector and the velocity vector.

1. Position Vector ($\vec{r}$): This vector points from the origin to the current location of the particle. If an object's position changes over time, we describe it as $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}$.
2. Velocity Vector ($\vec{v}$): This represents the rate of change of the position vector. Mathematically, it is the derivative of position with respect to time: $\vec{v} = \frac{d\vec{r}}{dt}$ In component form, this is $\vec{v} = v_x \hat{i} + v_y \hat{j}$.

The Particle Under Constant Velocity Model

When an object moves such that its velocity vector does not change (neither its speed nor its direction), we use the particle under constant velocity model. In this model, the displacement $\Delta \vec{r}$ over a time interval $\Delta t$ is simply: $\Delta \vec{r} = \vec{v} \Delta t$ This can be broken down into components: $\Delta x = v_x \Delta t$ and $\Delta y = v_y \Delta t$.

Vector Addition and Geometry

When multiple vectors act on an object (like two people pulling on a mule), we find the resultant vector through two-dimensional vector addition. The most efficient way to add vectors is to add their corresponding components: $\vec{A}_{total} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$

In cases where vectors do not form a right triangle, we cannot use the simple Pythagorean theorem to find the magnitude of the resultant. Instead, we must use the Law of Cosines: $c = \sqrt{a^2 + b^2 - 2ab \cos \theta}$ This allows us to find the third side of any triangle if we know the lengths of the other two sides and the angle between them.

Summary Checklist for Vector Problems:

• Identify the coordinate system (Cartesian or Polar).
• Break all vectors into $x$ and $y$ components.
• Perform your addition or subtraction using the components.
• Convert the final resultant back into magnitude and direction if required.