IB & Edexcel Physics Calculation Bootcamp | IB Edexcel 物理：计算题专项训练

📚 IB & Edexcel Physics Calculation Bootcamp | IB Edexcel 物理：计算题专项训练

Numerical problems form the backbone of IB and Edexcel Physics assessments, demanding not only conceptual understanding but also rigorous application of formulas, unit conversions, and significant figures. This article provides a structured bootcamp to sharpen your calculation skills, covering essential topics from kinematics to nuclear physics. By mastering techniques and avoiding common pitfalls, you will approach any calculation problem with confidence and precision.

计算题是 IB 和 Edexcel 物理考试的核心，它不仅要求对概念的理解，还要求严谨地应用公式、换算单位并正确处理有效数字。本文提供了一次系统的计算题专项训练，涵盖从运动学到核物理的基本主题。通过掌握技巧并避开常见陷阱，你将能够自信且精准地应对任何计算问题。

1. Understanding Units and Prefixes | 理解单位与词头

Always express quantities in SI base units before substituting into an equation. For example, convert 25 cm to 0.25 m, 72 km h⁻¹ to 20 m s⁻¹, and 300 g to 0.3 kg. Failure to do so is a leading cause of error in mechanics and electricity problems. Check that your final unit matches the quantity you are solving for – velocity should give m s⁻¹, not m s⁻².

在代入方程之前，务必将物理量用国际单位制的基本单位表示。例如，把 25 cm 换算成 0.25 m，72 km h⁻¹ 换算成 20 m s⁻¹，300 g 换算成 0.3 kg。不这样做是力学和电学问题出错的主要原因。最后还要检查得出的单位是否与你所求的物理量一致——速度应得到 m s⁻¹，而不是 m s⁻²。

Prefix	Symbol	Power of 10
nano	n	10⁻⁹
micro	μ	10⁻⁶
milli	m	10⁻³
centi	c	10⁻²
kilo	k	10³
mega	M	10⁶
giga	G	10⁹

When converting squared or cubed units, the power applies to both the number and the prefix – 1 cm³ is 1 × 10⁻⁶ m³, not 0.01 m³. Practise rewriting g cm⁻³ to kg m⁻³ by multiplying by 1000, as 1 g cm⁻³ = 1000 kg m⁻³.

当换算平方或立方单位时，幂次会同时作用在数字和词头上——1 cm³ 等于 1 × 10⁻⁶ m³，而不是 0.01 m³。练习将 g cm⁻³ 转换为 kg m⁻³ 时只需乘以 1000，因为 1 g cm⁻³ = 1000 kg m⁻³。

2. Significant Figures and Scientific Notation | 有效数字与科学计数法

IB and Edexcel mark schemes require final answers to reflect the precision of the data given. If the least precise value in the question has two significant figures, giving an answer to four significant figures will lose a mark. Count significant figures correctly: leading zeros are not significant (0.0032 has two s.f.), trailing zeros after a decimal point are significant (2.450 has four s.f.). Use scientific notation to avoid ambiguity – 1.5 × 10³ m is clearly two significant figures.

IB 和 Edexcel 的评分方案要求最终答案的精确度与题目所给数据一致。如果题目中最不精确的数值是两位有效数字，而你的答案给出了四位有效数字，就会失分。正确数出有效数字：前导零不算有效数字（0.0032 有两位），小数点后的末尾零是有效的（2.450 有四位）。使用科学计数法可以避免歧义——1.5 × 10³ m 一目了然是两位有效数字。

In multistep calculations, keep intermediate results to at least one extra significant figure, then round only the final answer. Be careful with logarithms: pH = 3.21 comes from [H⁺] with two significant figures in the mantissa, not the characteristic. For physics, apply this same rounding discipline consistently.

在多步计算中，中间结果至少要保留一位额外的有效数字，最后一步再对最终答案进行四舍五入。注意对数的有效数字规则：pH = 3.21 意味着 [H⁺] 的尾数部分有两位有效数字，而非首数。物理计算同样要始终遵守这一取舍规定。

3. Kinematics: The SUVAT Equations | 运动学：SUVAT 方程

The SUVAT equations apply only when acceleration is constant. Write down the five symbols: s, u, v, a, t. Identify three known values and the unknown you need. Choose the equation that excludes the remaining symbol. For example, if you do not know v, use s = u t + ½ a t².

SUVAT 方程仅适用于加速度恒定的情况。先写出五个物理量符号：s、u、v、a、t。确定三个已知量和需要求解的未知量，选择缺少第四个量的方程。例如，如果不知道 v，就使用 s = u t + ½ a t²。

v = u + a t

s = u t + ½ a t²

v² = u² + 2 a s

Pay attention to sign conventions. Define a positive direction – upwards or rightwards – and assign signs to u, v, a, and s accordingly. In free-fall problems near Earth’s surface, a = –9.81 m s⁻² if upward is positive. A common error is forgetting that displacement, velocity, and acceleration are vectors.

注意正负号规则。规定一个正方向——如向上或向右——然后相应地为 u、v、a、s 赋予正负号。在地表附近的自由落体问题中，若取向上为正，则 a = –9.81 m s⁻²。常见错误便是忘记位移、速度和加速度都是矢量。

4. Forces and Newton’s Laws | 力与牛顿定律

Start with a free-body diagram showing all forces acting on the object. Apply Newton’s second law in component form: ΣFₓ = m aₓ and ΣFᵧ = m aᵧ. Remember that the resultant force determines acceleration, not the individual forces. A block sliding down a frictionless incline accelerates with a = g sinθ, not g cosθ.

先画出受力示意图，标出作用在物体上的所有力。将牛顿第二定律写成分量形式：ΣFₓ = m aₓ 和 ΣFᵧ = m aᵧ。记住，是合外力决定加速度，而不是单个力。沿光滑斜面下滑的物块，加速度 a = g sinθ，而非 g cosθ。

For equilibrium problems, set both ΣFₓ = 0 and ΣFᵧ = 0. In connected-body systems (Atwood machine, pulleys), treat each mass separately and link their motions via a common string constraint: the magnitude of acceleration is the same, and tension appears with opposite signs in the equations.

对于平衡问题，令 ΣFₓ = 0 和 ΣFᵧ = 0。在连接体系统（阿特伍德机、滑轮）中，应隔离分析每个物体，并通过共同细绳的约束条件关联它们的运动：加速度大小相等，而张力在方程中符号相反。

Weight: W = m g, always acting downward.
重力：W = m g，始终竖直向下。
Normal contact force: perpendicular to surfaces.
法向接触力：垂直于接触面。
Friction: f ≤ μ R, with f = μ R when slipping.
摩擦力：f ≤ μ R，发生滑动时 f = μ R。

5. Work, Energy and Power | 功、能量与功率

The work done by a constant force is W = F d cosθ, where θ is the angle between the force and displacement. The area under a force–displacement graph gives the work done. Power is the rate of doing work: P = W / t or, for an object moving at constant velocity under a force parallel to motion, P = F v.

恒力做的功为 W = F d cosθ，其中 θ 是力与位移的夹角。力–位移曲线下的面积等于做功的大小。功率是做功的快慢：P = W / t，对于在平行力作用下匀速运动的物体，P = F v。

Energy must be conserved. Use the work–energy principle: net work equals change in kinetic energy, W_net = ½ m v² – ½ m u². For systems with conservative forces, total mechanical energy (kinetic + potential) remains constant if no dissipative forces act. Include elastic potential energy ½ k x² when springs are present.

能量必须守恒。运用功能原理：合外力做功等于动能的变化量，W_net = ½ m v² – ½ m u²。对于只受保守力作用的系统，若无耗散力，机械能（动能+势能）守恒。如果存在弹簧，要计入弹性势能 ½ k x²。

Be careful with efficiency: η = (useful power output) / (total power input). Often expressed as a percentage. A motor lifting a mass converts electrical energy into gravitational potential energy with some energy lost to heat.

注意效率计算：η = (有用输出功率) / (总输入功率)。常用百分数表示。电动机提升重物时，将电能转化为重力势能，并有一部分能量以热量形式耗散。

6. Circular Motion and Gravitation | 圆周运动与引力

An object moving uniformly in a circle experiences a centripetal acceleration a_c = v² / r = ω² r directed toward the centre. The centripetal force is F_c = m v² / r. This force is not a new kind of force – it is provided by tension, gravity, friction, or the normal reaction. Identify the force that points to the centre and equate it to m v² / r.

作匀速圆周运动的物体具有始终指向圆心的向心加速度 a_c = v² / r = ω² r。向心力 F_c = m v² / r。向心力并非一种新的力——它由拉力、重力、摩擦力或法向反力提供。找出指向圆心的力，并令其等于 m v² / r。

v = ω r

For gravitational fields, Newton’s law of gravitation F_g = G M m / r² gives the force between two point masses. The gravitational field strength at a distance r from a mass M is g = G M / r². When a satellite orbits, set gravitational force equal to centripetal force: G M m / r² = m v² / r, which leads to v = √(G M / r).

在引力场中，牛顿万有引力定律 F_g = G M m / r² 给出两个质点间的引力。距离质量 M 为 r 处的引力场强度为 g = G M / r²。卫星环绕运行时，令万有引力等于向心力：G M m / r² = m v² / r，由此导出 v = √(G M / r)。

7. Electric Fields and Potential | 电场与电势

The electric field strength due to a point charge Q is E = k Q / r², where k = 1/(4π ε₀). The force on a test charge q is F = q E. Work done moving a charge through a potential difference ΔV is W = q ΔV. Always watch the sign of the charge – negative charges move opposite to the field direction.

点电荷 Q 产生的电场强度为 E = k Q / r²，其中 k = 1/(4π ε₀)。检验电荷 q 所受的力为 F = q E。将电荷移动通过电势差 ΔV 所做的功为 W = q ΔV。务必留意电荷的正负——负电荷运动方向与电场方向相反。

In a uniform electric field between parallel plates, E = ΔV / d. This relationship is used to find the force on a charge in Millikan’s oil-drop experiment or to determine the trajectory of charged particles in oscilloscopes. Equate electrical force to weight for a suspended charged particle: q E = m g.

在平行板间的匀强电场中，E = ΔV / d。这一关系可用于密立根油滴实验中求电荷所受的力，或确定示波器中带电粒子的轨迹。对于悬浮的带电微粒，使电场力与重力平衡：q E = m g。

8. Circuits and Internal Resistance | 电路与内阻

Ohm’s law V = I R applies to ohmic conductors at constant temperature. For a complete circuit, the terminal potential difference is V = ε – I r, where ε is the electromotive force (emf) and r the internal resistance of the cell. The short-circuit current occurs when R = 0, giving I_max = ε / r. Practise plotting V against I: the gradient is –r and the y-intercept is ε.

欧姆定律 V = I R 适用于温度恒定的欧姆导体。对整个回路而言，路端电压为 V = ε – I r，其中 ε 为电动势，r 为电源内阻。当外电阻 R = 0 时发生短路，此时电流 I_max = ε / r。练习绘制 V-I 图线：斜率为 –r，纵截距为 ε。

Resistors in series: R_total = R₁ + R₂ + … . Resistors in parallel: 1/R_total = 1/R₁ + 1/R₂ + … . Use the potential divider formula for series circuits: V₁ = V_total × R₁/(R₁ + R₂). For parallel circuits, all branches share the same voltage. Kirchhoff’s current law: sum of currents entering a junction equals sum of currents leaving. Kirchhoff’s voltage law: sum of emfs equals sum of p.d.s around any closed loop.

串联电阻：R_total = R₁ + R₂ + … 。并联电阻：1/R_total = 1/R₁ + 1/R₂ + … 。串联电路使用分压公式：V₁ = V_total × R₁/(R₁ + R₂)。并联电路中各支路电压相等。基尔霍夫电流定律：流入节点的电流之和等于流出节点的电流之和。基尔霍夫电压定律：沿任意闭合回路，电动势之和等于各段电势差之和。

9. Radioactive Decay Calculations | 放射性衰变计算

Radioactive decay follows an exponential law: N = N₀ e^(–λ t), where N₀ is the initial number of nuclei, λ the decay constant, and t the time elapsed. The activity A = λ N decays similarly: A = A₀ e^(–λ t). The half-life t₁⸝₂ is related to λ by t₁⸝₂ = ln 2 / λ. You may be given a table of count rates and asked to determine λ or t₁⸝₂ by plotting a graph of ln A against t.

放射性衰变遵循指数规律：N = N₀ e^(–λ t)，其中 N₀ 为初始核素数目，λ 为衰变常量，t 为经过的时间。活度 A = λ N 也以同样的规律衰变：A = A₀ e^(–λ t)。半衰期 t₁⸝₂ 与 λ 的关系为 t₁⸝₂ = ln 2 / λ。题目可能会给出一组计数率数据，要求通过绘制 ln A 对 t 的图线来求出 λ 或 t₁⸝₂。

Be careful with units: λ has units of s⁻¹, yr⁻¹, etc., and must be consistent with the time unit. When dealing with mass, remember that the number of atoms N = (m / molar mass) × Avogadro’s number. For decay chains or successive products, focus on the parent nucleus unless instructed otherwise.

注意单位：λ 的单位为 s⁻¹、年⁻¹ 等，必须与时间单位协调。涉及质量时，记住原子数目 N = (m / 摩尔质量) × 阿伏伽德罗常数。对于衰变链或子体产物，除非另有说明，否则应重点关注母核。

10. Exam Technique and Common Pitfalls | 考试技巧与常见陷阱

Read each question twice. Underline the command term: ‘Calculate’, ‘Determine’, ‘Estimate’. Show all steps of your working – even if the final answer is wrong, you can earn marks for correct physics. Write the formula first, substitute numbers with units, then solve algebraically before picking up the calculator.

每题读两遍。圈出指令词：“计算”、“确定”、“估计”。写出所有解题步骤——即便最终答案错误，你也能因正确的物理过程而得分。先写公式，代入带单位的数值，然后在用计算器之前先进行代数求解。

Common mistake: using the wrong mass or acceleration in connected systems. Always redraw free-body diagrams for each body. Another error: forgetting to convert hours to seconds, or litres to cubic metres. Keep a list of standard form conversions handy.

常见错误：在连接体问题中使用了错误的质量或加速度。一定要为每个物体重新画出受力示意图。另一个错误：忘记把小时换算成秒，或把升换成立方米。牢记一张标准单位换算表。

Check that your answer is physically reasonable – e.g. a car’s speed cannot exceed a few hundred metres per second.
检查答案在物理上是否合理——例如汽车的速度不会超过每秒几百米。
If you finish early, re-calculate values using a different method to verify.
如果提前做完，用不同的方法重新计算以作验证。

For graphical analysis, use at least half the graph paper, label axes with quantities and units, and draw a line of best fit. The gradient and intercept must be determined from the line, not from raw data points. When calculating gradient, use a large triangle to minimise error.

进行图像分析时，至少要用到半张坐标纸，坐标轴标明物理量和单位，并画出最佳拟合线。斜率和截距必须从拟合线上获取，而不能直接用原始数据点。计算斜率时，选取一个大的三角形以减小误差。

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IB & Edexcel Physics Calculation Bootcamp | IB Edexcel 物理：计算题专项训练