Analyzing Motion with Graphs

The Two Fundamental Kinematic Graphs

In kinematics, almost everything revolves around two types of graphs. A common mistake students make is confusing them, so always check your axes first!

1. Distance-Time Graphs: Show how far an object has travelled from a starting point over time.
2. Speed-Time Graphs: Show how an object's speed changes over time.

Let's break down how to read each one.

1. Interpreting Distance-Time Graphs

On a distance-time graph, the y-axis is distance and the x-axis is time.

• Gradient = Speed: The most important rule for these graphs is that the gradient of the line represents the object's speed. $$\text{Gradient}=\frac{\text{change in }y}{\text{change in }x}=\frac{\text{change in Distance}}{\text{change in Time}}=\text{Speed}$$
• Horizontal Line: A gradient of zero means the speed is zero. The object is stationary.
• Straight, Sloping Line: A constant, non-zero gradient means the object is moving at a constant speed. The steeper the line, the faster it's moving.
• Curved Line: A changing gradient means the speed is changing (i.e., the object is accelerating or decelerating).

Worked Example: Distance-Time Graph

Let's look at a typical exam problem.

Analysis of the problem:

• Part (a) Average Speed: To find the average speed for the journey to the office, we need the total distance traveled and the total time taken for that part of the journey.
  • From the graph, the total distance to the office is 28 km.
  • The total time taken is 10:30 - 08:30 = 2 hours.
  • Average Speed = Total Distance / Total Time = 28 km / 2 hours = 14 km/h.
• Part (b) Completing the graph: Kai returns home at a constant speed of 21 km/h.
  • Distance to travel = 28 km.
  • Time = Distance / Speed = 28 km / 21 km/h = 4/3 hours.
  • 4/3 hours = 1 hour and 1/3 of an hour = 1 hour and 20 minutes.
  • He leaves the office at 17:30. His arrival time is 17:30 + 1h 20m = 18:50.
  • To complete the graph, we draw a straight line from the point (17:30, 28 km) down to (18:50, 0 km).

2. Interpreting Speed-Time Graphs

Speed-time graphs are more common in exams and contain more information. The y-axis is speed, and the x-axis is time. There are two key rules to remember.

1. Gradient = Acceleration:

   $$\text{Gradient}=\frac{\text{change in }y}{\text{change in }x}=\frac{\text{change in Speed}}{\text{change in Time}}=\text{Acceleration}$$
   • Positive gradient: Positive acceleration (speeding up).
   • Negative gradient: Negative acceleration, or deceleration (slowing down).
   • Zero gradient (horizontal line): Zero acceleration, meaning constant speed.

2. Area under the Graph = Distance Travelled:

   • If you multiply Speed (y-axis) by Time (x-axis), you get Distance. Therefore, the area under the graph represents the total distance.
   • You can almost always find this area by splitting it into simple shapes: triangles, rectangles, and trapeziums.

Let's consolidate these two rules with a clear, focused resource.

Now, let's see these rules applied in a video walkthrough of some typical exam questions.

Advanced Problems: Working Backwards

A common, more challenging question gives you the total distance and asks you to find a missing time or speed. This requires you to form an algebraic equation using the 'Area = Distance' rule.

Your Task:
Before looking at the solution in the image, try to solve this yourself.

1. Recognise the shapes: The area under the graph can be split into a trapezium and a triangle.
2. Write expressions for the area:
   • Area of Trapezium (0 to 24s): $\frac{1}{2}(a+b)h=\frac{1}{2}(v+v+10)\times 24$
   • Area of Triangle (24 to 40s): $\frac{1}{2}bh=\frac{1}{2}\times 16\times (v+10)$
3. Form an equation: The sum of these two areas is the total distance. Crucially, check the units! The distance is given as 1.24 km, which must be converted to 1240 m. $$\frac{1}{2}(2v+10)\times 24+\frac{1}{2}\times 16\times (v+10)=1240$$
4. Solve the equation for $v$.

Compare your method and final answer with the one provided in the image. This type of question is excellent for picking up method marks, even if you make a small calculation error.

Estimating with Curved Graphs

While most IGCSE questions involve straight lines, you may be asked to interpret curved speed-time graphs.

• Instantaneous Acceleration: The acceleration at a single point in time is found by drawing a tangent to the curve at that point and calculating its gradient.
• Estimating Distance: The distance travelled is still the area under the curve. You can estimate this by splitting the area into several trapeziums of equal width.

Drawing tangents accurately takes practice. The key is to have your ruler just "kiss" the curve at the required point, with the gap between the line and the curve looking symmetrical on both sides of the point.

Conclusion

You've now covered the essential skills for solving kinematics problems using graphs. By remembering a few core rules, you can turn these questions into a reliable source of marks.

Key Takeaways:

• Always check the axes to determine if you have a distance-time or speed-time graph.
• For distance-time graphs, the key relationship is Gradient = Speed.
• For speed-time graphs, there are two key relationships:
  • Gradient = Acceleration
  • Area under graph = Distance
• For area calculations, split the graph into rectangles, triangles, and trapeziums.
• For algebraic problems, form an equation by setting the total area equal to the given total distance.
• For curved graphs, use tangents to find instantaneous rates and trapeziums to estimate area.

Preview of the Next Lesson:
In our next lesson, we move to a new module: "Geometric Transformations and Vectors". We'll start by learning how to describe and perform single geometric transformations: rotation, reflection, translation, and enlargement. This builds on what you learned about transforming function graphs, but applies the concepts to 2D shapes on a coordinate grid.