📚 IB Physics: EE Planning Sheet 2024 Formula Derivations | IB 物理：EE Planning Sheet 2024 公式推导
Beginning your IB Physics Extended Essay means you will spend a good deal of time on the planning sheet, and one of the most valuable sections is the formula derivation. Whether your investigation is experimental, data-based, or purely theoretical, showing a clear, logical progression from fundamental principles to the equations you actually use strengthens your EE enormously. This guide helps you design and present formula derivations that meet the expectations of the 2024 planning sheet, ensuring your mathematical work is rigorous, well documented, and closely tied to your research question.
开始你的 IB 物理扩展论文（EE）意味着你会在规划表上投入大量时间，而其中一个最有价值的部分就是公式推导。无论你进行的是实验性、基于数据还是纯理论的研究，清晰地展示从基本原理到你实际使用方程式的逻辑推导过程都能极大地强化你的 EE。本指南将帮助你在 2024 年规划表中设计和呈现公式推导，确保你的数学处理严谨、记录充分，并与你的研究问题紧密相连。
1. The Role of the EE Planning Sheet | 规划表的作用
The 2024 EE planning sheet is not just a formality; it is a working document where you outline your research question, methodology, and theoretical framework. The derivation section asks you to show the mathematical backbone behind any key formula you will use in data analysis or theoretical predictions. Supervisors and examiners use this to judge whether you truly understand the physics, not merely plugging numbers into a textbook equation.
2024 年 EE 规划表并非流于形式，而是一份工作文件，用于概述你的研究问题、方法和理论框架。其中推导部分要求你展示任何将用于数据分析或理论预测的关键公式背后的数学主干。导师和考官以此来评判你是否真正理解了物理，而不仅仅是把数字代入教科书公式。
When you complete your planning sheet early, the derivation serves as a checkpoint: if you cannot derive a formula from first principles, you probably need to deepen your understanding before embarking on data collection. It also allows you to spot unnecessary complexity or wrong assumptions right at the outset, saving time later.
当你及早完成规划表时，推导就成为一个检查点：如果你无法从第一原理推导出某个公式，很可能你需要在开展数据收集之前加深理解。它还能让你在项目一开始就发现不必要的复杂之处或错误的假设，从而节省后续的时间。
2. Deciding Where Derivations Belong | 确定推导在论文中的位置
In the planning sheet, the derivation may appear under the ‘Theoretical background’ or ‘Methodology’ heading. It is not an appendix to the final essay but rather a preview of how you will connect theory to your work. Decide early whether your formula derivation will underpin a hypothesis, explain a trend in your data, or justify a specific calculation method.
在规划表中，推导可能出现在“理论背景”或“方法论”标题下。它不是最终论文的附录，而是你将如何把理论与你的工作联系起来的预览。尽早决定你的公式推导是支撑一个假设、解释数据趋势，还是证明某种计算方法的合理性。
For an experimental EE, the derivation often shows how a measured quantity relates to the variables you control. For a database or simulation-based EE, it proves that your modelled relationship is physically sound. The planning sheet should reflect this connection by indicating which part of the methodology the derived formula will serve.
对于实验性 EE，推导常常展示被测量量与你所控制变量之间的关系。对于基于数据库或模拟的 EE，它证明你所建模的关系在物理上是成立的。规划表中应反映出这种联系，指明推导出的公式将服务于方法论的哪一个部分。
3. Selecting a Research Question That Invites Derivation | 选择适合推导的研究问题
Not all research questions are equally suited to formula derivation. A question like ‘How does the period of a simple pendulum depend on its length?’ is rich in derivation paths, while a purely comparative study may offer little scope. Aim for a question where a theoretical model can be constructed from established laws (Newton’s laws, conservation of energy, Maxwell’s equations, etc.) and then refined through the derivation.
并不是所有研究问题都同样适合公式推导。像“单摆的周期如何取决于其长度？”这样的问题拥有丰富的推导路径，而纯粹的对比研究可能几乎没有推导的空间。要选择这样一个问题：你可以从已知定律（牛顿定律、能量守恒、麦克斯韦方程组等）出发构建一个理论模型，然后通过推导对其进行细化。
For 2024, IB encourages you to link your derivation to the assessment criteria, especially Criterion C (Critical thinking). A question that allows you to derive an equation, examine its limits, and discuss approximations naturally fulfils this. When drafting your planning sheet, list the primary physical law and state how your derivation will modify or apply it.
2024 年，IB 鼓励你将推导与评估标准联系起来，特别是标准 C（批判性思维）。一个允许你推导方程、检验其局限性并讨论近似条件的问题，自然会满足这一要求。在起草规划表时，列出你使用的主要物理定律，并说明你的推导将如何修正或应用它。
4. Gathering the Fundamental Equations | 收集基本方程
Every derivation must start from a set of accepted equations—these are your axioms. For example, if you are deriving the range equation for a projectile, you begin with Newton’s second law and the kinematic equations. In your planning sheet, state these starting equations clearly and cite their source. Typical IB Physics starting points include F = ma, conservation of momentum, the ideal gas law PV = nRT, or Ohm’s law V = IR.
每一个推导都必须从一组公认的方程开始——这些是你的公理。例如，如果你正在推导抛体的射程方程，你将从牛顿第二定律和运动学方程出发。在规划表中，清晰地写明这些初始方程并注明来源。典型的 IB 物理起点包括 F = ma、动量守恒、理想气体定律 PV = nRT 或欧姆定律 V = IR。
Use the planning sheet to define every symbol in these equations. Write them in a table to avoid ambiguity. This not only makes your derivation easier to follow but also helps you spot missing variables.
在规划表中定义这些方程中的每个符号。用表格将它们列出来可以避免歧义。这不仅让你的推导更易于理解，还能帮助你发现缺失的变量。
| Symbol | Meaning | Unit |
| F | Net force | N |
| m | Mass | kg |
Such a table in your planning sheet is a small touch that examiners notice.
规划表里这样一张小小的表格，会让考官眼前一亮。
5. Building the Derivation from First Principles | 从第一原理构建推导
A strong derivation walks the reader through every logical step, never jumping to a conclusion without justification. Start with the fundamental equation, apply the conditions specific to your research question, and isolate the variable of interest. For instance, if you are deriving the velocity of a rolling object at the bottom of an incline, you might begin with conservation of energy: ½mv² + ½Iω² = mgh. Then substitute ω = v/r and solve for v.
一个有力的推导会引导读者走过每一个逻辑步骤，绝不越过证明就跳到结论。从基本方程出发，应用你的研究问题特定的条件，然后分离出感兴趣的变量。例如，如果你要推导一个滚动物体在斜面底部的速度，你可以从能量守恒开始：½mv² + ½Iω² = mgh。然后代入 ω = v/r 并求解 v。
v = √(2gh / (1 + I/(mr²)))
The planning sheet should show such steps in a clear, numbered sequence. Even if the final EE presents only the key steps, the planning sheet benefits from showing the full reasoning to demonstrate depth of understanding.
规划表应以清晰的编号序列展示这些步骤。即使最终的 EE 只呈现关键步骤，规划表最好能展示完整推理以体现理解的深度。
6. Performing Step-by-Step Algebraic Manipulations | 进行一步步代数操作
Algebraic errors are a common reason for lost marks on an EE. In your planning sheet, show intermediate simplifications explicitly. If you cross-multiply, factor, or cancel terms, write those actions as separate lines. This habit will help you later when you check the derivation against experimental data, because you can trace any discrepancy back to a specific algebraic move.
代数错误是 EE 丢分的一个常见原因。在规划表中，明确展示中间的简化过程。如果你做交叉相乘、因式分解或消去项，将这些操作写成单独的行。这个习惯在你日后对照实验数据检查推导时会很有帮助，因为你能将任何偏差追溯到一个具体的代数操作上。
Use the following format in your planning sheet to keep derivations tidy:
- State the assumption or algebraic rule applied (e.g., ‘Divide both sides by t’).
- Show the resulting expression.
- Indicate the next goal (e.g., ‘Now isolate a’).
在规划表中采用以下格式来保持推导整洁：
- 说明所应用的假设或代数规则（例如，“两边同除以 t”）。
- 展示得到的表达式。
- 指出下一个目标（例如，“现在分离出 a”）。
7. Incorporating Calculus and Vector Calculus | 纳入微积分与矢量微积分
Many IB Physics EEs are enhanced by calculus-based derivations, such as finding induced emf from Faraday’s law or deriving the kinetic energy expression from work. If your research question warrants calculus, your planning sheet should show the differential equations set up and their integration. For instance, to derive the velocity of a falling object under air resistance, you might start with m dv/dt = mg – kv.
许多 IB 物理 EE 借助基于微积分的推导得到强化，比如用法拉第定律求感应电动势，或者从功推导动能表达式。如果你的研究问题适合微积分，规划表应展示微分方程的建立及其积分过程。例如，要推导受空气阻力下落物体的速度，你可以从 m dv/dt = mg – kv 开始。
v(t) = (mg/k)(1 – e⁻ᵏᵗ/ᵐ)
In the planning sheet, clearly note any boundary conditions you apply, such as v(0) = 0. Do not assume that the reader will infer them. If you are working with vectors, show the resolution into components and keep the direction convention consistent throughout.
在规划表中，清楚地注明你应用的任何边界条件，例如 v(0) = 0。不要假设读者会自行推断它们。如果你处理的是矢量，展示分量分解并在整个推导过程中保持方向约定的一致性。
8. Common Pitfalls in IB Physics Derivations | IB 物理推导中的常见误区
A frequent mistake is to begin a derivation with the very equation you are trying to prove. For example, starting with the period formula T = 2π√(L/g) and then rearranging it does not constitute a derivation; you must begin with Newton’s laws or energy considerations. Your planning sheet should guard against circular reasoning by explicitly naming the starting principle.
一个常见错误是从你正要证明的方程式本身开始推导。例如，以周期公式 T = 2π√(L/g) 为起点然后重新排列，这不构成推导；你必须从牛顿定律或能量角度出发。你的规划表应通过明确指定起始原理来防止循环论证。
Another pitfall is ignoring the validity range of an approximation. If you use the small-angle approximation sinθ ≈ θ for a pendulum, state the condition (θ < 10°) and discuss how it affects your final result. Examiners reward candidates who recognise the limits of their own model.
另一个误区是忽略近似的有效范围。如果你对单摆使用了小角度近似 sinθ ≈ θ，要说明条件（θ < 10°）并讨论它对最终结果的影响。考官会欣赏那些认识到自己模型局限性的考生。
9. Presenting Derivations with Visual Clarity | 以视觉清晰的方式呈现推导
The planning sheet is a working document, but it still needs to be legible. Use bullet points, numbered lines, and clear separation between steps. A well-structured derivation might look like this:
规划表是一份工作文件，但它仍然需要清晰易读。使用项目符号、编号行，并在步骤之间保持清晰的间隔。一个结构清晰的推导可能看起来像这样：
(1) Starting equation: a = F/m
(2) Substitute F = qE for a charged particle: a = qE/m
(3) Use kinematic equation v = u + at with u = 0: v = (qE/m)t
In the planning sheet, each step can be accompanied by a brief English commentary, which later becomes the prose in your final essay.
在规划表中，每一步都可以附上简短的英文注释，这些注释日后将成为你最终论文中的论述文字。
10. Linking the Derivation to Experimental Variables | 将推导与实验变量联系起来
A derivation is only valuable if it produces an equation that can be tested. After you obtain the final formula, rewrite it in a linear form suitable for graphing. For example, if you derive T² = (4π²/g) L, you can state that a graph of T² versus L should yield a straight line with slope 4π²/g. This direct link to data analysis should appear in the planning sheet.
只有当推导产出一个可以检验的方程时，它才具有价值。在得到最终公式后，将其改写为适合绘制图线的线性形式。例如，如果你推导出 T² = (4π²/g) L，你可以指出 T² 对 L 的图线应是一条斜率为 4π²/g 的直线。这种与数据分析的直接联系应该出现在规划表中。
The planning sheet should also explain which variables you will measure and how the derived relationship allows you to determine a physical constant, such as g, from the slope. This shows the examiner that the derivation is not an isolated theoretical exercise but a core part of your investigation.
规划表还应解释你将测量哪些变量，以及推导出的关系如何让你从斜率确定一个物理常数，比如 g。这向考官表明推导不是一个孤立的理论练习，而是你研究的核心部分。
11. Reviewing and Validating the Derivation | 审查与验证推导
Before submitting your planning sheet, test your derivation with simple, known cases. If you substitute extreme values, does the equation behave as physics predicts? For example, if your derived formula for terminal velocity depends on mass, check that as mass → 0, the velocity → 0, or as mass → ∞, the velocity tends to a finite limit. Document these sanity checks in the planning sheet.
在提交规划表之前，用简单的已知案例检验你的推导。如果你代入极值，方程式是否如物理所预测的那样？例如，如果你推导的终极速度公式依赖于质量，检查当质量 → 0 时，速度 → 0，或当质量 → ∞ 时，速度趋近于一个有限极限。将这些合理性检查记录在规划表中。
If possible, run a preliminary numerical simulation in a spreadsheet and compare the results with your derived equation. Any unexpected behaviour can be investigated and explained, which deepens the critical thinking element of your EE.
如果可能，在电子表格中运行一个初步的数值模拟，并将结果与你的推导方程进行比较。任何出乎意料的行为都可以被调查和解释，这将加深你 EE 中的批判性思维元素。
12. Finalising the Derivation Section for the Planning Sheet | 为规划表提交最终推导部分
Compile your derivation into a self-contained subsection of the planning sheet that includes the starting assumptions, a concise logical sequence, and the final equation in both raw and linearised forms. Add a brief sentence on how the derivation answers your research question. This summary helps both you and your supervisor see the complete story.
将你的推导整理到规划表的一个独立子部分中，包含起始假设、简洁的逻辑序列以及原始形式和线性化形式的最终方程。添加一句简短的话，说明推导如何回答你的研究问题。这份总结有助于你和你的导师看清整个脉络。
Remember that the 2024 planning sheet is an opportunity to plan meticulously. A well-crafted derivation section gives you a strong foundation and builds confidence. It also makes writing the final essay much smoother, because the theoretical framework is already solid.
请记住，2024 年规划表是一个让你进行周密规划的机会。一个精心打磨的推导部分能为你奠定坚实的基础并建立信心。它也会让最终的论文写作顺畅得多，因为理论框架已经牢固了。
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