Welcome to Kinematics Graphs!
Ever tried to describe a car journey to a friend? You could tell them the speeds and times, but showing them a picture—a graph—is much more powerful. In this chapter, we will learn how to turn motion into "pictures" using graphs. Whether you're aiming for an A* or just trying to get your head around the basics, these notes will help you master how to read and draw these graphs with confidence.
1. The Foundation: Displacement-Time Graphs
A displacement-time graph shows where an object is (relative to a starting point) at any given moment. Usually, displacement (s) is on the vertical y-axis, and time (t) is on the horizontal x-axis.
What the lines tell us:
• A horizontal line: This means the displacement isn't changing. The object is stationary (sitting still).
• A straight sloping line: This means the displacement is changing at a steady rate. The object is moving with constant velocity.
• A curved line: The "steepness" is changing, which means the velocity is changing (the object is accelerating or decelerating).
The Secret Key: The Gradient
The most important thing to remember is: The gradient of a displacement-time graph is the velocity.
\( \text{Gradient} = \frac{\text{Change in } s}{\text{Change in } t} = \text{Velocity} \)
Did you know? If the slope goes "downhill" (negative gradient), it means the object is moving back toward its starting position or in the opposite direction!
Quick Review:
Flat line = Stopped
Steep line = Fast velocity
Gentle slope = Slow velocity
Key Takeaway: To find how fast something is moving on this graph, just find the slope!
2. The Powerhouse: Velocity-Time Graphs
This is the most common graph you’ll see in Mechanics. Velocity (v) is on the y-axis and time (t) is on the x-axis. It contains two pieces of hidden treasure: the gradient and the area.
Finding Acceleration (The Gradient)
Just like before, the "steepness" tells us something. The gradient of a velocity-time graph is the acceleration.
\( \text{Acceleration} (a) = \frac{\text{Change in } v}{\text{Change in } t} \)
• A positive gradient means the object is speeding up (accelerating).
• A negative gradient means the object is slowing down (decelerating).
• A horizontal line means the velocity is constant (acceleration is zero).
Finding Distance and Displacement (The Area)
This is where it gets interesting! The area under a velocity-time graph represents the displacement.
• To find total distance, you add up all the areas (even the parts below the x-axis).
• To find displacement, you treat areas above the x-axis as positive and areas below the x-axis as negative.
Analogy: Imagine a velocity-time graph for a runner. The area is like the "footprints" they left behind. The more area under the line, the further they have run.
Step-by-Step: Finding the Area
1. Break the shape under the graph into simple shapes: rectangles and trapeziums (or triangles).
2. Use the formulas: \( \text{Area of Rectangle} = \text{base} \times \text{height} \) and \( \text{Area of Trapezium} = \frac{1}{2}(a+b)h \).
3. Add them together to get the total displacement.
Key Takeaway: Gradient = Acceleration; Area = Displacement.
3. Acceleration-Time Graphs
These graphs show how acceleration changes over time. In many A Level problems, you will see horizontal lines here because we often deal with constant acceleration.
The Secret Key: The Area
The area under an acceleration-time graph represents the change in velocity.
If the acceleration is \( 5 \, \text{ms}^{-2} \) for \( 3 \) seconds, the area is \( 5 \times 3 = 15 \). This means the object is now going \( 15 \, \text{ms}^{-1} \) faster than it was at the start.
Don't worry if this seems tricky at first... we rarely look at the gradient of this graph (which is called "jerk"), so just focus on what the area tells you!
Key Takeaway: Area = Change in velocity.
4. Common Mistakes to Avoid
1. Mixing up Distance and Displacement: On a velocity-time graph, if a car goes forward and then reverses, the displacement might be zero (it's back where it started), but the distance is the total ground covered. Always check if the question asks for distance or displacement!
2. Forgetting Units: Always label your gradients and areas with the correct units (e.g., \( \text{ms}^{-1} \) for velocity, \( \text{ms}^{-2} \) for acceleration).
3. Reading the Wrong Axis: Before you calculate anything, look at the y-axis. Is it Displacement? Velocity? Acceleration? Using the wrong rule for the wrong graph is the most common error.
5. Memory Aid: The "G.A.V.A." Rule
Use this simple chain to remember which "tool" to use for each graph:
Displacement Graph
• Gradient \( \rightarrow \) Velocity
Velocity Graph
• Gradient \( \rightarrow \) Acceleration
• Area \( \rightarrow \) Displacement
Acceleration Graph
• Area \( \rightarrow \) Change in Velocity
Summary Table for Quick Reference
Graph Type | Gradient represents... | Area represents...
Displacement-Time | Velocity | (Not used)
Velocity-Time | Acceleration | Displacement
Acceleration-Time | (Not used) | Change in Velocity
Final Tip: When drawing these graphs, always use a ruler for straight lines and clearly mark the values on the axes where the motion changes. Clear graphs lead to clear marks!