Graphical Representation of Motion (Position-Time Graphs)

TL;DR

Position-time graphs show an object's location over time, with the slope indicating its velocity. A straight line means constant velocity, while a curved line indicates changing velocity (acceleration). By analyzing the graph's shape and slope, you can understand an object's motion.

1. The Mental Model

Imagine you're watching a car drive by and you're drawing its path on a piece of paper, marking where it is at every second. A position-time graph is exactly that: a visual story of an object's location over time, making it easy to see if it's moving, standing still, speeding up, or slowing down.

2. The Core Material

Graphical representation is a powerful way to visualize the motion of a point, body, or particle. We often look at different types of graphs, like displacement-time, velocity-time, or acceleration-time, but we'll focus now on position-time graphs (which are also called displacement-time graphs, as displacement is a change in position).

Interpreting Position-Time Graphs

A position-time graph plots the position (or displacement, 's') of an object on the vertical (y) axis against time ('t') on the horizontal (x) axis.

The key to understanding these graphs is the slope. The slope of a position-time graph tells you the object's velocity. Remember, velocity is the rate of change of displacement with respect to time.
$$ \text{Velocity (v)} = \frac{\text{Displacement (d)}}{\text{Time (t)}} $$
In the context of a graph, the slope is rise/run, which is change in position / change in time.

Let's look at different scenarios:

a) Object at Rest (Not Moving)

If an object is not moving, its position doesn't change over time.
- Graph Appearance: A horizontal line.
- Interpretation from Source: "From the graph, it is clear that with the passage of time, there is no change in the position of the body, it remains at point A."
- Slope: Zero. This means the velocity is zero.

b) Uniform Motion (Constant Positive Velocity)

If an object is in uniform motion, it covers equal distances (or displacements) in equal intervals of time. This means its velocity is constant.
- Graph Appearance: A straight line with a positive slope.
- Interpretation from Source: "From the graph, it is clear that in equal intervals of time, the body covers equal distances, so the motion is uniform and graph is a straight line."
- Slope: Constant and positive. This represents a constant positive velocity.

c) Non-Uniform Motion (Changing Velocity/Acceleration)

When an object's velocity is changing, its motion is non-uniform. This means it's accelerating or decelerating.
- Graph Appearance: A curved line.
- Increasing Speed (Positive Slope, becoming steeper): The object covers more and more distance in equal time intervals. "with the passage of time, the body is covering more and more distance in equal time i.e., the speed of the body is increasing. Hence, the slope of graph is positive."
- Decreasing Speed (Positive Slope, becoming flatter): The object covers less and less distance in equal time intervals. "with the passage of time, the body is covering lesser and lesser distance in equal time i.e., the speed of the body is decreasing. Hence, the slope of the graph is negative." (Note: The source says "negative," but if the position is still increasing, the slope is positive, just decreasing.) The key is that the rate of change of position is slowing down.
- Slope: Changing (either increasing or decreasing). This indicates acceleration.

graph TD
    A["Position-Time Graph"] --> B{"Line Shape"}
    B -- "Horizontal" --> C["Object at Rest"]
    C --> D["Position constant"]
    C --> E["Slope = 0"]
    C --> F["Velocity = 0"]

    B -- "Straight Line (Slanted)" --> G["Uniform Motion"]
    G --> H["Equal displacement in A["equal time"]
    G --> I["Constant Slope"]
    G --> J["Constant Velocity"]

    B -- "Curved Line" --> K["Non-Uniform Motion"]
    K --> L["Unequal displacement in equal time"]
    K --> M["Changing Slope"]
    K --> N["Changing Velocity (Acceleration/Deceleration)"]

3. Worked Example

Let's say you have a position-time graph for a toy car.

• From t = 0s to t = 2s: The line is straight and goes from position 0m to 10m.
  • Here, the slope is (10m - 0m) / (2s - 0s) = 5 m/s. This means the car is moving at a uniform positive velocity of 5 m/s.
• From t = 2s to t = 4s: The line is horizontal at position 10m.
  • The slope is (10m - 10m) / (4s - 2s) = 0 m/s. This means the car is at rest, not moving.
• From t = 4s to t = 6s: The line is straight and goes from position 10m back to 0m.
  • The slope is (0m - 10m) / (6s - 4s) = -10m / 2s = -5 m/s. This means the car is moving at a uniform negative velocity of -5 m/s (moving in the opposite direction).

4. Key Takeaways

• The vertical axis of a position-time graph represents the object's position or displacement.
• The horizontal axis of a position-time graph represents time.
• The slope of a position-time graph gives you the object's velocity.
• A horizontal line on a position-time graph means the object is at rest (zero velocity).
• A straight, slanted line means uniform motion (constant velocity).
• A curved line on a position-time graph means non-uniform motion (changing velocity, i.e., acceleration).
• A positive slope indicates movement in a positive direction, while a negative slope indicates movement in a negative direction.

Common Mistakes to Avoid:
- Don't confuse position with velocity; the slope is velocity, not the y-value itself.
- Don't assume a curved line always means speeding up; it could be speeding up or slowing down depending on how the curve bends.
- Remember that "negative slope" means moving backward (or in the opposite direction), not necessarily "slowing down."
- Forgetting that "uniform motion" on this graph means constant velocity, not necessarily zero velocity.

5. Now Try It

Sketch three different position-time graphs on paper: one for an object moving at a constant speed away from an origin, one for an object standing still, and one for an object slowing down as it approaches the origin. For each graph, label the axes and briefly explain what the slope tells you about the object's motion at different points. You'll know you're successful if your sketches clearly differentiate these three types of motion using the shape of the line and if your explanations correctly link the slope to the velocity.