Motion Graphs

TL;DR

Motion graphs visually represent an object's position, velocity, or acceleration over time. Interpreting their slopes and areas helps you understand how motion changes. You'll typically encounter position-time, velocity-time, and acceleration-time graphs, each revealing different aspects of motion.

1. The Mental Model

Think of motion graphs as stories about movement. Each point on a graph tells you "where" or "how fast" an object is at a specific "when." By looking at the shape and steepness of the lines, you can quickly see if an object is standing still, moving steadily, speeding up, or slowing down.

2. The Core Material

Motion graphs are powerful tools in physics because they allow you to visualize and analyze movement without complex calculations. There are three main types you'll focus on:

Position-Time Graphs (x-t graphs)

Photo by Sergey Meshkov on Pexels

These graphs show an object's position (often denoted as 'x' or 'd' for distance/displacement) on the y-axis against time ('t') on the x-axis.
* Slope: The slope of a position-time graph tells you the object's velocity.
* A horizontal line means the object is stationary (velocity = 0).
* A straight, upward-sloping line means constant positive velocity.
* A straight, downward-sloping line means constant negative velocity (moving in the opposite direction).
* A curved line means changing velocity, which implies acceleration.
* Y-intercept: This tells you the object's starting position.

Velocity-Time Graphs (v-t graphs)

Photo by AlphaTradeZone on Pexels

These graphs show an object's velocity ('v') on the y-axis against time ('t') on the x-axis.
* Slope: The slope of a velocity-time graph tells you the object's acceleration.
* A horizontal line means constant velocity (acceleration = 0).
* A straight, upward-sloping line means constant positive acceleration (speeding up in the positive direction or slowing down in the negative direction).
* A straight, downward-sloping line means constant negative acceleration (slowing down in the positive direction or speeding up in the negative direction).
* A curved line means changing acceleration.
* Area Under the Curve: The area between the line and the time axis on a velocity-time graph tells you the object's displacement (change in position).
* Area above the axis is positive displacement.
* Area below the axis is negative displacement.

Acceleration-Time Graphs (a-t graphs)

Photo by AlphaTradeZone on Pexels

These graphs show an object's acceleration ('a') on the y-axis against time ('t') on the x-axis.
* Area Under the Curve: The area between the line and the time axis on an acceleration-time graph tells you the change in velocity.
* Slope: The slope of an acceleration-time graph is called "jerk" and isn't usually covered at this level.

Here's how these graphs relate to each other:

graph TD
    A["Position-Time Graph (x-t)"] --> B{"Slope gives Velocity"};
    B --> C["Velocity-Time Graph (v-t)"];
    C --> D{"Slope gives Acceleration"};
    C --> E{"Area gives Displacement"};
    D --> F["Acceleration-Time Graph (a-t)"];
    F --> G{"Area gives Change in Velocity"};

3. Worked Example

Let's analyze a simple velocity-time graph for a car:

Imagine a car's velocity-time graph shows:
1. From t=0s to t=5s, the velocity goes from 0 m/s to 10 m/s in a straight line.
2. From t=5s to t=10s, the velocity stays constant at 10 m/s.
3. From t=10s to t=12s, the velocity drops from 10 m/s to 0 m/s in a straight line.

Let's find the acceleration and displacement for each segment:

Segment 1 (0s to 5s):
* Acceleration (slope): (Change in velocity) / (Change in time) = (10 m/s - 0 m/s) / (5s - 0s) = 10 m/s / 5s = 2 m/s² (constant positive acceleration).
* Displacement (area): This is a triangle. Area = 0.5 * base * height = 0.5 * (5s) * (10 m/s) = 25 m.

Segment 2 (5s to 10s):
* Acceleration (slope): (10 m/s - 10 m/s) / (10s - 5s) = 0 m/s / 5s = 0 m/s² (constant velocity, no acceleration).
* Displacement (area): This is a rectangle. Area = base * height = (10s - 5s) * (10 m/s) = 5s * 10 m/s = 50 m.

Segment 3 (10s to 12s):
* Acceleration (slope): (0 m/s - 10 m/s) / (12s - 10s) = -10 m/s / 2s = -5 m/s² (constant negative acceleration, slowing down).
* Displacement (area): This is a triangle. Area = 0.5 * base * height = 0.5 * (12s - 10s) * (10 m/s) = 0.5 * 2s * 10 m/s = 10 m.

Total Displacement: 25m + 50m + 10m = 85m.

4. Key Takeaways

• Position-time graph slope = velocity.
• Velocity-time graph slope = acceleration.
• Velocity-time graph area = displacement.
• Acceleration-time graph area = change in velocity.
• A horizontal line on a position-time graph means the object is at rest.
• A horizontal line on a velocity-time graph means constant velocity (zero acceleration).
• Curved lines on x-t or v-t graphs indicate changing slopes, meaning acceleration or changing acceleration, respectively.

Common Mistakes to Avoid:
- Confusing graph types: Always check the axes labels (x-t, v-t, a-t) before interpreting a graph.
- Mixing up slope and area: Remember which graph type uses slope for one quantity and area for another.
- Ignoring negative values: Displacement or velocity can be negative, indicating movement in the opposite direction.
- Misinterpreting horizontal lines: A horizontal line means constant whatever is on the y-axis. So, constant position, constant velocity, or constant acceleration.
- Assuming all motion is constant: Don't assume constant velocity or acceleration unless the graph is a straight line.

5. Now Try It

Sketch a velocity-time graph for the following scenario: A ball is dropped from rest, speeds up as it falls, hits the ground, bounces up to half its original drop height, and then falls again.

What to do: Draw the axes, label them (velocity on y, time on x). Then, draw a series of lines or curves representing the ball's velocity over time for each phase of its motion (dropping, hitting, bouncing up, falling again). Think about when velocity is positive, negative, or zero, and when acceleration is involved.

What success looks like: Your graph should show an initial increasing negative velocity (falling), an instantaneous change in velocity upon impact (a jump from negative to positive velocity), decreasing positive velocity (bouncing up), reaching zero velocity at the peak of its bounce, and then increasing negative velocity again (falling). The slopes should indicate constant acceleration due to gravity (a straight line for each falling/rising segment).