Motion in 2 dimensions

Welcome to Motion in 2 Dimensions!

In your previous studies, you likely looked at objects moving in a straight line—up and down or left and right. In this chapter, we take things to the next level: Motion in 2 Dimensions. Imagine a bird flying through the air or a footballer kicking a ball across a pitch; these objects are moving horizontally and vertically at the same time. Don't worry if this seems a bit more complex at first! The secret to mastering 2D motion is realizing that it is just two 1D problems happening simultaneously, linked together by time.

1. The Language of 2D Kinematics

When we move from one dimension to two, we use vectors to describe where an object is and how it is moving. Instead of a single number, we use components (usually i for the horizontal and j for the vertical).

Key Terms:
Position Vector (\(\mathbf{r}\)): This tells us where an object is relative to a fixed origin. It is usually written as \(\mathbf{r} = x\mathbf{i} + y\mathbf{j}\).
Displacement (\(\mathbf{s}\)): The change in the position vector. If you start at \(\mathbf{r}_1\) and end at \(\mathbf{r}_2\), your displacement is \(\mathbf{s} = \mathbf{r}_2 - \mathbf{r}_1\).
Distance: This is a scalar. It is the actual length of the path traveled. To find the distance from the origin, we find the magnitude of the position vector: \(|\mathbf{r}| = \sqrt{x^2 + y^2}\).
Velocity (\(\mathbf{v}\)): A vector describing the rate of change of displacement.
Speed: This is the magnitude of the velocity vector: \(Speed = |\mathbf{v}| = \sqrt{v_x^2 + v_y^2}\).

Analogy: Think of a position vector like a GPS coordinate. The 'i' tells the GPS how far East/West to go, and the 'j' tells it how far North/South. Together, they pinpoint exactly where you are on the map.

Quick Review:
1. Vectors have magnitude and direction.
2. Average Velocity = \(\frac{Total\ Displacement}{Total\ Time}\)
3. Average Speed = \(\frac{Total\ Distance}{Total\ Time}\)

2. Constant Acceleration in 2D (SUVAT)

If an object's acceleration is constant (it doesn't change over time), we can use our familiar SUVAT equations. The beauty of vectors is that the equations look exactly the same as they did in 1D, just with bold letters to represent vectors!

\(\mathbf{v} = \mathbf{u} + \mathbf{a}t\)
\(\mathbf{s} = \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2\)
\(\mathbf{s} = \frac{1}{2}(\mathbf{u} + \mathbf{v})t\)
\(\mathbf{r} = \mathbf{r}_0 + \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2\) (where \(\mathbf{r}_0\) is the starting position)

The Golden Rule: You can treat the i and j components completely separately. Solve the horizontal world and the vertical world individually. The only thing they share is time (\(t\)).

Memory Aid: "T is the Tie"
Time is the only variable that is exactly the same for both the horizontal and vertical components of motion. If you are stuck, try to find \(t\) in one dimension to use it in the other!

Key Takeaway: Always separate your vector information into a horizontal list and a vertical list before you start calculating.

3. Calculus in Kinematics

What if the acceleration is not constant? This is where calculus comes in. Just like in 1D, we can differentiate to find "rates of change" and integrate to find "accumulated change." We just do it for each component.

To go "Forward" (Differentiate):
Position (\(\mathbf{r}\)) \(\rightarrow\) Velocity (\(\mathbf{v}\)) \(\rightarrow\) Acceleration (\(\mathbf{a}\))
\(\mathbf{v} = \frac{d\mathbf{r}}{dt}\)
\(\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{r}}{dt^2}\)

To go "Backward" (Integrate):
Acceleration (\(\mathbf{a}\)) \(\rightarrow\) Velocity (\(\mathbf{v}\)) \(\rightarrow\) Position (\(\mathbf{r}\))
\(\mathbf{v} = \int \mathbf{a}\ dt\)
\(\mathbf{r} = \int \mathbf{v}\ dt\)

Common Mistake to Avoid: When integrating, don't forget the constant of integration! In 2D, this constant will be a vector (e.g., \(\mathbf{c} = c_1\mathbf{i} + c_2\mathbf{j}\)). You usually find this using the "initial conditions" (where the object was at \(t=0\)).

4. Finding the Path (Cartesian Equations)

Sometimes, we don't want to know where an object is at a specific time; we just want to see the shape of its path on a graph. This is called the Cartesian Equation, and it only involves \(x\) and \(y\), not \(t\).

Step-by-Step: How to find the path
1. Write down the expressions for \(x\) and \(y\) in terms of \(t\) (these come from your position vector).
2. Rearrange the simplest equation (usually the \(x\) equation) to make \(t\) the subject.
3. Substitute this expression for \(t\) into the other equation.
4. Simplify to get an equation in the form \(y = f(x)\).

Example: If \(x = 2t\) and \(y = t^2\), then \(t = \frac{x}{2}\). Substituting this into \(y\) gives \(y = (\frac{x}{2})^2\), or \(y = \frac{1}{4}x^2\). This tells us the object is moving in a parabola!

5. Relative Position and Problem Solving

In 2D problems, you are often asked about the relationship between two different particles, \(A\) and \(B\).

Relative Position: The position of \(B\) relative to \(A\) is given by \(\mathbf{r}_B - \mathbf{r}_A\).
Analogy: If you are standing at point A, the relative position vector is the "arrow" you would have to draw to point directly at B.

Important Connections:
1. Direction of Motion: The direction a particle is moving is always the direction of its velocity vector (\(\mathbf{v}\)).
2. Direction of Force: According to Newton's Second Law (\(\mathbf{F}=m\mathbf{a}\)), the resultant force always acts in the same direction as the acceleration vector (\(\mathbf{a}\)).

Did you know?
If the velocity vector and acceleration vector are perpendicular (at 90 degrees), the object will change direction but its speed will stay exactly the same! This is how satellites stay in circular orbits around the Earth.

Final Chapter Summary

- 2D motion uses vectors: Always keep your i and j components distinct.
- Use SUVAT for constant acceleration: Treat horizontal and vertical separately, linked by time \(t\).
- Use Calculus for variable acceleration: Differentiate to move from position to acceleration; integrate to go back.
- Cartesian Equations: Eliminate \(t\) to find the shape of the path (\(y\) in terms of \(x\)).
- Relative Position: Subtract the "observer's" vector from the "target's" vector (\(\mathbf{r}_{target} - \mathbf{r}_{observer}\)).