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AQA GCSE · Question 21.2 · Geometry and Measures
Work out an estimate for the total distance covered by the ball.
Work out an estimate for the total distance covered by the ball.
How to approach this question
1. Understand that the distance covered is the area under the velocity-time graph.
2. Split the graph into simpler shapes. A good way is to split it at t=4 and t=10. This gives you a curved section, a rectangle, and a triangle.
3. Calculate the area of the rectangle and the triangle using standard formulas.
4. Estimate the area of the curved section. You can do this by approximating it as a trapezium or by counting squares.
5. Add the areas of the three sections together to get the total estimated distance.
Full Answer
The total distance covered is the area under the velocity-time graph. We can split the area into three parts: a curved section (0 to 4s), a rectangle (4s to 10s), and a triangle (10s to 16s).
**Step 1: Area of the rectangle (Area B, from t=4 to t=10).**
The graph shows a constant velocity of 8 m/s between t=4 and t=10.
Width = 10 - 4 = 6 s
Height = 8 m/s
Area B = width × height = 6 × 8 = 48 m.
**Step 2: Area of the triangle (Area C, from t=10 to t=16).**
Base = 16 - 10 = 6 s
Height = 8 m/s
Area C = 0.5 × base × height = 0.5 × 6 × 8 = 24 m.
**Step 3: Estimate the area of the curved section (Area A, from t=0 to t=4).**
We can approximate this area by treating it as a trapezium.
The parallel sides are the velocities at t=0 (which is 0) and t=4 (which is 8). The height of the trapezium is the time interval (4).
Area A ≈ 0.5 × (a + b) × h = 0.5 × (0 + 8) × 4 = 0.5 × 8 × 4 = 16 m.
(Alternatively, one could count squares or use the trapezium rule with more strips for better accuracy).
**Step 4: Calculate the total distance.**
Total Distance = Area A + Area B + Area C
Total Distance ≈ 16 + 48 + 24 = 88 m.
**Answer: Approximately 88 m.** (Answers may vary slightly depending on the method used for the curved section, e.g., 86m to 90m).
The total distance covered by the ball is equal to the area under the velocity-time graph. We can split the graph into three sections from t=0 to t=16.
**Section 1: t = 0 to t = 4 (Curved section)**
This part is a curve. We can estimate its area by treating it as a trapezium. The "height" of the trapezium is the time interval (4 - 0 = 4). The parallel sides are the velocities at t=0 and t=4.
Velocity at t=0 is 0 m/s.
Velocity at t=4 is 8 m/s.
Area₁ ≈ ½ × (sum of parallel sides) × height
Area₁ ≈ ½ × (0 + 8) × 4 = 16 m.
**Section 2: t = 4 to t = 10 (Rectangle)**
This is a rectangle with width (10 - 4 = 6) and height 8.
Area₂ = width × height = 6 × 8 = 48 m.
**Section 3: t = 10 to t = 16 (Triangle)**
This is a triangle with base (16 - 10 = 6) and height 8.
Area₃ = ½ × base × height = ½ × 6 × 8 = 24 m.
**Total Distance:**
Total Distance = Area₁ + Area₂ + Area₃
Total Distance ≈ 16 + 48 + 24 = 88 m.
Common mistakes
✗ Forgetting to divide by 2 when calculating the area of the triangle.
✗ Incorrectly calculating the width of the rectangle or the base of the triangle.
✗ Using an incorrect method to estimate the area under the curve.
✗ Reading values incorrectly from the graph axes.
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