Parametric modelling in motion and design
| English | 中文 | Pinyin |
|---|---|---|
| parametric model/ˌpærəˈmetrɪk ˈmɒdl/ | 参数模型 | cān shù mó xíng |
When does a mathematical path stop describing a real flight?
- A ball’s height and horizontal position depend on the same time. A Cartesian path cannot tell us when it is reached without the time model.
- This lesson studies parametric model 参数模型: A model expressing two quantities using a common variable and stated assumptions.
Choose the mathematical structure
- Name the parameter, its units and allowed interval. Evaluate both model quantities at the same parameter value. Eliminate the parameter to relate the quantities, then interpret endpoints and limitations. A mathematical curve beyond the allowed interval need not describe the physical situation.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines parametric model?
A model expressing two quantities using a common variable and stated assumptions.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
A simplified flight model uses x=6t and y=20t−5t² in metres, with t in seconds and 0≤t≤4. It assumes constant horizontal speed and downward acceleration 10 metres per second squared, ignoring air resistance. The noninitial ground time is t=4, giving x=24 m. Completing the square gives y=20−5(t−2)², so maximum height is 20 m at t=2. Eliminating t gives y=10x/3−5x²/36 for 0≤x≤24. For a rectangle of width t and length t+2 metres, P=4t+4 and A=t(t+2). With 0<t≤5, A=(P²−16)/16 and 4<P≤24.
Parametric modelling in motion and design
Name the parameter, its units and allowed interval
Identify which statements preserve the original parameter, curve segment and contextual assumptions.
Find the horizontal ground distance in the stated flight model.
y=t(20−5t)=0 gives t=4 after launch; x=6×4=24 m.
Test a tempting shortcut
- The flight formula giving negative height after t=4 does not describe continued free flight below the ground. Do not use t measured in minutes with coefficients defined for seconds. In the rectangle model, t=0 is excluded because the width must be positive.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The Cartesian flight curve is a physically valid model for every real x. This claim is false. Explain which definition or assumption it violates.
Find the maximum height in the same model.
Complete the square: y=20−5(t−2)², with maximum 20 m.
The Cartesian flight curve is a physically valid model for every real x.
The flight formula giving negative height after t=4 does not describe continued free flight below the ground. Do not use t measured in minutes with coefficients defined for seconds. In the rectangle model, t=0 is excluded because the width must be positive.
Interpret a new situation
- Check a model against observed data and its assumptions. Wind or air resistance could require refinement of the flight model; a different length–width relation changes the rectangle model. The acceleration 10 here is a stated simplification, not a replacement for a paper’s instructed value of g.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the rectangle area when its width t is 5 metres.
A=t(t+2)=5×7=35 m².
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · C. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A model expressing two quantities using a common variable and stated assumptions. Choose the relationship, show the method, check its assumptions and interpret the result.