Introduction to One-Dimensional Motion

TL;DR

One-dimensional motion describes an object's movement along a straight line, defined by its position, velocity, and acceleration. You'll learn to describe where an object is, how fast it's moving, and how its speed changes over time. Understanding these basic concepts lets you predict future motion or reconstruct past movement.

1. The Mental Model

Imagine a car driving on a perfectly straight road. You're trying to describe its journey: where it starts, where it ends up, how fast it goes, and whether it speeds up or slows down. That's one-dimensional motion.

2. The Core Material

When we talk about one-dimensional motion, we're focusing on movement along a single axis, like the x-axis. Everything is relative to an origin, a zero point you choose.

2.1 Position

Your position tells you exactly where an object is along that line. We usually use 'x' for position. If you're at x = 5 meters, you're 5 meters away from the origin in the positive direction. If you're at x = -3 meters, you're 3 meters away in the negative direction. It's a vector quantity, meaning it has both magnitude (how far) and direction (which way from the origin).

2.2 Displacement

Displacement is the change in an object's position. It's the straight-line distance from your starting point to your ending point, and it includes direction. We often write it as $\Delta x = x_{final} - x_{initial}$. If you walk 5 meters forward and then 3 meters backward, your total displacement is 2 meters forward, not 8 meters total distance walked.

2.3 Velocity

Velocity tells you how fast an object is moving AND in what direction. It's the rate of change of position.
* Average velocity is your total displacement divided by the time it took: $v_{avg} = \Delta x / \Delta t$.
* Instantaneous velocity is your velocity at a precise moment in time, like what your speedometer reads.

Speed is just the magnitude of velocity – how fast without the direction.

2.4 Acceleration

Acceleration is the rate at which your velocity changes. If your velocity is increasing, you're accelerating. If it's decreasing (slowing down), you're also accelerating, but in the opposite direction of your motion (sometimes called deceleration).
* Average acceleration is the change in velocity divided by the time taken: $a_{avg} = \Delta v / \Delta t$.
* Instantaneous acceleration is the acceleration at a specific moment.

Think of it like this: if you're on a bike and pedaling harder, you're accelerating. If you hit the brakes, you're accelerating in the opposite direction. If you're moving at a constant speed, your acceleration is zero.

Here's how these concepts build on each other:

graph TD
    A["Time (t)"] --> B["Position (x)"]
    B --> C["Change in Position (Δx)"]
    C --> D["Velocity (v = Δx/Δt)"]
    D --> E["Change in Velocity (Δv)"]
    E --> F["Acceleration (a = Δv/Δt)"]

3. Worked Example

Let's say a car starts at position $x_0 = 0$ m. It travels to $x_1 = 20$ m in 5 seconds, then turns around and travels back to $x_2 = 5$ m in another 3 seconds.

1. Total time: $5 \text{ s} + 3 \text{ s} = 8 \text{ s}$.
2. Initial position: $x_{initial} = 0 \text{ m}$.
3. Final position: $x_{final} = 5 \text{ m}$.
4. Total displacement: $\Delta x = x_{final} - x_{initial} = 5 \text{ m} - 0 \text{ m} = 5 \text{ m}$.
5. Average velocity: $v_{avg} = \Delta x / \Delta t = 5 \text{ m} / 8 \text{ s} = 0.625 \text{ m/s}$.

Notice that even though the car traveled a total distance of $20 \text{ m} + 15 \text{ m} = 35 \text{ m}$, its displacement was only $5 \text{ m}$. That's because displacement only cares about the start and end points.

4. Key Takeaways

• Position tells you an object's location relative to an origin.
• Displacement is the change in position, including direction, not total distance.
• Velocity is the rate of change of position, including direction (speed + direction).
• Acceleration is the rate of change of velocity.
• A positive value often means "to the right" or "up," while negative means "to the left" or "down," based on your chosen coordinate system.

Common mistakes to avoid:
- Confusing distance with displacement.
- Confusing speed with velocity.
- Forgetting that acceleration can be negative even if an object is moving forward (it just means it's slowing down).
- Not choosing a consistent positive direction for your calculations.

5. Now Try It

Imagine you're walking along a straight path. You start at the 0 meter mark. You walk 10 meters forward in 4 seconds. Then, you walk 2 meters backward in 2 seconds.

Calculate your:
1. Final position.
2. Total displacement.
3. Average velocity for the entire walk.

Success means you've correctly identified the final position, accounted for the change in direction for displacement, and used the total time for average velocity.