Welcome to the Story of Motion!
In this chapter, we are going to look at Graphical Representation within the world of Mechanics. Think of a graph as a "picture" of a journey. Instead of reading a long list of numbers, we can look at a single shape and immediately see if an object is speeding up, slowing down, or standing perfectly still.
Don't worry if you find graphs a bit intimidating at first. We are going to break them down into two main types: Displacement-Time graphs and Velocity-Time graphs. By the end of these notes, you'll be able to "read" a journey just by looking at the lines!
1. Displacement-Time (\(s-t\)) Graphs
A displacement-time graph shows how far an object is from its starting point as time goes by. Displacement (\(s\)) is always on the vertical \(y\)-axis, and Time (\(t\)) is always on the horizontal \(x\)-axis.
What the lines tell us:
A Straight Diagonal Line: This means the object is moving at a constant velocity. The steeper the line, the faster it is moving.
A Horizontal Line: This means the displacement isn't changing. The object is stationary (stopped).
A Curved Line: This means the velocity is changing. The object is either accelerating (getting steeper) or decelerating (getting flatter).
The Secret Ingredient: The Gradient
In a displacement-time graph, the gradient (the slope) is the most important thing to remember.
Gradient = Velocity
\( \text{Velocity} = \frac{\text{change in displacement}}{\text{change in time}} \)
Analogy: Imagine you are walking to a shop. If you walk at a steady pace, your "story" is a straight diagonal line. if you stop to chat with a friend, your line goes flat (time continues, but you don't move!). If you realize the shop is closing and start to run, your line curves upwards steeply!
Quick Review: Displacement-Time
1. Gradient = Velocity.
2. Flat line = Stopped.
3. Straight slope = Constant speed.
2. Velocity-Time (\(v-t\)) Graphs
A velocity-time graph shows how fast an object is moving at any given second. Velocity (\(v\)) is on the vertical axis, and Time (\(t\)) is on the horizontal axis.
What the lines tell us:
A Horizontal Line: Warning! This does not mean the object is stopped. It means the velocity is staying the same. The object is moving at a constant speed.
A Straight Diagonal Line: This shows the velocity is changing at a steady rate. This is constant acceleration.
A Line on the \(x\)-axis (where \(v=0\)): This is the only time the object is actually stationary.
The Two Big Secrets of \(v-t\) Graphs
Velocity-time graphs give us two pieces of "hidden" information:
1. The Gradient = Acceleration
If the line goes up, the gradient is positive (speeding up). If the line goes down, the gradient is negative (slowing down/decelerating).
2. The Area Under the Graph = Displacement
If you calculate the area between the line and the time axis, you have found the total distance travelled (displacement).
Did you know? You can find the displacement of a complex journey just by breaking the area under the graph into simple shapes like rectangles and triangles!
Quick Review: Velocity-Time
1. Gradient = Acceleration.
2. Area Under Graph = Displacement.
3. Flat line = Constant speed (not stopped!).
3. Summary Table & Memory Aids
It can be easy to mix these up! Use this simple trick to remember what the gradient represents as you move "up" the levels of kinematics:
The "G-V-A" Trick:
Moving from Displacement to Velocity? Find the Gradient.
Moving from Velocity to Acceleration? Find the Gradient.
Key Takeaway Summary:
Displacement-Time Graph:
- Gradient = Velocity (\(v\))
Velocity-Time Graph:
- Gradient = Acceleration (\(a\))
- Area = Displacement (\(s\))
4. Common Mistakes to Avoid
Mistake 1: Confusing "Stationary" on different graphs.
On an \(s-t\) graph, a horizontal line means "stopped." On a \(v-t\) graph, a horizontal line means "moving at a steady speed." Always check your axes before you decide what the line means!
Mistake 2: Forgetting units.
Always ensure your time is in seconds (s) and your displacement is in metres (m). If the question gives you minutes or kilometers, convert them first!
Mistake 3: Negative gradients.
A negative gradient on a displacement-time graph means the object is moving backwards toward the start. A negative gradient on a velocity-time graph means the object is slowing down.
5. Step-by-Step: Finding Displacement from a \(v-t\) Graph
If you are asked to find the total distance travelled from a velocity-time graph, follow these steps:
Step 1: Identify the shapes under the graph. Look for triangles (for acceleration/deceleration) and rectangles (for constant speed).
Step 2: Calculate the area of each shape.
- Area of rectangle = \(base \times height\)
- Area of triangle = \(\frac{1}{2} \times base \times height\)
Step 3: Add all the areas together. This total is your total displacement!
Encouraging Note: If the shapes look weird, don't panic! Most exam questions use a combination of simple triangles and rectangles. Just split them up one by one.
Final Key Points Summary
- Displacement-time (\(s-t\)): Steepness shows how fast you are moving.
- Velocity-time (\(v-t\)): Steepness shows how fast you are speeding up; the space underneath shows how far you've gone.
- Constant acceleration: Represented by a straight diagonal line on a velocity-time graph.