📚 IB Physics: Core Formulas and Question Types for Work, Energy and Power | IB物理：功、能量与功率核心公式及题型归纳
Work, energy and power form one of the most frequently tested areas in IB Physics. This topic connects mechanics, circular motion and thermal physics, and it appears in both Paper 1 and Paper 2. In this article, we will summarise the core formulas, clarify common misconceptions, and classify typical exam question types to help you revise efficiently.
功、能量与功率是IB物理中考查频率最高的板块之一。它连接了力学、圆周运动与热学，在Paper 1和Paper 2中都会出现。本文将为同学们梳理核心公式、澄清常见误区，并按题型归类整理典型考题，帮助大家高效复习。
1. Work Done: Definition and Formula | 功的定义与公式
Work is done when a force acts on an object and the object moves in the direction of the force. The general equation is:
W = F × s × cos θ
Here, W is work measured in joules (J), F is the applied force in newtons (N), s is the displacement in metres (m), and θ is the angle between the force and displacement vectors.
当力作用在物体上，且物体沿力的方向发生位移时，力就对物体做了功。一般公式为：
W = F × s × cos θ
其中 W 是功，单位焦耳（J）；F 是力，单位牛顿（N）；s 是位移，单位米（m）；θ 是力与位移方向之间的夹角。
rmal force and gravity do no work when an object slides horizontally along a flat surface.
学习提示：当力与位移平行时，θ = 0°，W = Fs；当力与位移垂直时，W = 0。例如，物体在水平面上滑动时，支持力和重力都不做功。
2. Kinetic Energy and the Work–Energy Theorem | 动能与动能定理
Kinetic energy is the energy an object possesses due to its motion. The formula is:
Eₖ = ½ m v²
where m is mass and v is speed. Kinetic energy is a scalar quantity and is always positive or zero.
动能是物体由于运动而具有的能量，公式为：
Eₖ = ½ m v²
其中 m 是质量，v 是速度大小。动能是标量，永远大于或等于零。
The work–energy theorem states that the net work done on an object equals its change in kinetic energy:
Wnet = ΔEₖ = ½ m v² − ½ m u²
This theorem is extremely useful for solving problems involving variable forces or curved paths, because it avoids detailed kinematic analysis.
动能定理指出：合外力对物体所做的功等于物体动能的变化量：
Wnet = ΔEₖ = ½ m v² − ½ m u²
这个定理在解决变力或曲线运动问题时非常有用，因为可以绕过复杂的运动学分析。
3. Gravitational Potential Energy | 重力势能
Near the Earth's surface, gravitational potential energy is given by:
Eₚ = m g h
where g is gravitational field strength (about 9.8 N kg⁻¹ on Earth, but IB often uses 10 N kg⁻¹ for simplicity), and h is the height above a chosen reference level.
在地球表面附近，重力势能公式为：
Eₚ = m g h
其中 g 是重力场强度（地球上约9.8 N kg⁻¹，IB考试中常取10 N kg⁻¹），h 是相对于所选参考平面的高度。
Important: The reference level is arbitrary. What matters in calculations is the change in height, Δh, not the absolute value of h. When solving free-fall or pendulum questions, always define your zero level clearly at the start.
注意：参考平面可以任意选择。计算中真正重要的是高度变化Δh，而不是h的绝对值。在解自由落体或单摆问题时，务必先明确设定零势能面。
4. Elastic Potential Energy and Springs | 弹性势能与弹簧
For an ideal spring obeying Hooke's law, the elastic potential energy stored is:
Eₑ = ½ k x²
where k is the spring constant and x is the extension or compression from the natural length.
对于遵循胡克定律的理想弹簧，储存的弹性势能为：
Eₑ = ½ k x²
其中 k 是劲度系数，x 是相对原长的伸长量或压缩量。
ergy stored is ½kx². The factor of ½ appears because the force increases linearly from zero to kx during the stretching process.
考试中要注意：弹簧弹力是F = kx，但储存的能量是½kx²。系数½的出现是因为在拉伸过程中力从零线性增加到kx。
5. Power: Average and Instantaneous | 功率：平均与瞬时
Power is the rate at which energy is transferred or work is done. The average power is:
P = W / t = ΔE / t
where P is measured in watts (W = J s⁻¹).
功率是能量转移或做功的快慢。平均功率为：
P = W / t = ΔE / t
其中 P 的单位是瓦特（W = J s⁻¹）。
When a constant force acts on an object moving at constant velocity, instantaneous power can also be calculated as:
P = F × v
This form is commonly used for vehicle problems, where engine power is constant and the driving force decreases as speed increases.
当恒力作用于匀速运动的物体时，瞬时功率还可以表示为：
P = F × v
这个形式常用于交通工具问题：当发动机功率恒定时，速度增大则牵引力减小。
6. Efficiency in Energy Transfers | 能量转化中的效率
Efficiency is the ratio of useful output energy (or power) to total input energy (or power):
η = (useful output energy / total input energy) × 100%
or equivalently using power: η = Pout / Pin × 100%.
效率是有用输出能量（或功率）与总输入能量（或功率）之比：
η = (有用输出能量 / 总输入能量) × 100%
也可以用功率表示为：η = Pout / Pin × 100%。
In IB questions, efficiency is often combined with energy conservation. For example, an electric motor may transfer 80% of electrical energy into kinetic energy and the rest into thermal energy. Make sure you identify which energy is 'useful' in the context of the question.
在IB题目中，效率常与能量守恒结合。例如，电动机将80%的电能转化为动能，其余转化为热能。解题时要根据题目情境判断哪种能量是“有用的”。
7. Conservation of Mechanical Energy | 机械能守恒
When only conservative forces (such as gravity and ideal spring forces) do work, the total mechanical energy remains constant:
Eₖ + Eₚ = constant
This is a powerful tool for solving roller-coaster, pendulum, projectile and spring problems.
当只有保守力（如重力、理想弹簧弹力）做功时，系统总机械能保持不变：
Eₖ + Eₚ = constant
这是解决过山车、单摆、抛体和弹簧问题的重要工具。
If non-conservative forces (like friction or air resistance) are present, mechanical energy is not conserved. In such cases, use:
E_initial + Wnon-conservative = Efinal
where the work done by friction is negative.
如果存在非保守力（如摩擦力或空气阻力），机械能不守恒。此时应使用：
E_initial + Wnon-conservative = Efinal
其中摩擦力做功为负值。
8. Common Question Type 1: Work Done by a Constant Force | 题型一：恒力做功
In this type, you are given a force, a displacement and an angle. The key is to correctly resolve the force component along the displacement.
此类题目会给出力、位移和夹角。关键在于正确分解力在位移方向上的分量。
that the vertical component of the force does no work because displacement is horizontal.
示例：小孩用50 N的力拉雪橇，拉力方向与水平面成30°角，水平位移20 m。做功 = 50 × 20 × cos 30° ≈ 866 J。注意拉力的竖直分量不做功，因为位移是水平的。
9. Common Question Type 2: Energy Conservation in Vertical Motion | 题型二：竖直运动中的能量守恒
These problems often involve a mass sliding down a frictionless incline or falling from a height. You can equate the initial potential energy to the final kinetic energy.
这类问题通常涉及物体沿光滑斜面下滑或从高处落下。可将初始势能等于末态动能。
: A ball of mass 2 kg is dropped from a height of 5 m. Just before hitting the ground, v = √2gh = √2 × 9.8 × 5 = 9.9 m s⁻¹. You can also use kinematics to check, but energy is often faster.
典型思路：一个2 kg的小球从5 m高处自由下落。落地前瞬间速度 v = √2gh = √2 × 9.8 × 5 ≈ 9.9 m s⁻¹。也可以用运动学公式验证，但能量法通常更快。
10. Common Question Type 3: Power and Variable Force | 题型三：功率与变力问题
Car and motor problems typically give a constant power output P, and you must find the maximum speed when the driving force equals the resistive force.
汽车或电动机问题通常给定恒定输出功率P，需要求当牵引力等于阻力时的最大速度。
= 20000 / 400 = 50 m s⁻¹. At speeds lower than this, the excess power accelerates the car.
示例：一辆1000 kg的汽车受到恒定的400 N阻力。若发动机输出20 kW，则最大速度 v = P / F = 20000 / 400 = 50 m s⁻¹。当车速低于此值时，多余功率用于加速。
11. Common Question Type 4: Work Done by Friction and Thermal Energy | 题型四：摩擦力做功与热能
When an object slides along a rough surface, the work done against friction is converted into thermal energy. This is a common Paper 2 extended-response context.
物体沿粗糙表面滑动时，克服摩擦力做的功转化为热能。这是Paper 2简答题的常见情境。
riction = μ × N, where μ is the coefficient of kinetic friction and N is the normal force.
摩擦力 f = μ × N，其中μ为动摩擦因数，N为正压力。
Work done by friction = f × s × cos 180° = −f × s.
摩擦力做功 = f × s × cos 180° = −f × s。
The thermal energy generated = f × s (positive value).
产生的热能等于f × s（取正值）。
surface with μ = 0.2, mass 5 kg. The thermal energy produced = 0.2 × 5 × 9.8 × 3 = 29.4 J.
例如，一个5 kg的物块在μ = 0.2的粗糙面上滑行3 m。产生的热能 = 0.2 × 5 × 9.8 × 3 = 29.4 J。
12. Common Question Type 5: Graphs and Area Interpretation | 题型五：图像与面积含义
In IB Physics, force–displacement graphs are used to represent work. The area under a force–displacement graph equals the work done by the force.
在IB物理中，力-位移图像用于表示做功。力-位移图线下方的面积等于力所做的功。
Similarly, the area under a power–time graph represents energy transferred. You should be comfortable calculating areas for rectangles, triangles and trapeziums.
同理，功率-时间图线下方的面积表示转移的能量。你需要熟练计算矩形、三角形和梯形的面积。
Another common graph is the potential energy versus position curve. The slope of an Eₚ– F = −dEₚ/dx. This is an AIHL/AAHL friendly concept but also appears in SL energy contexts.
另一个常见图像是势能-位置曲线。Eₚ–x图像的斜率等于力的负值：F = −dEₚ/dx。虽然这常与数学HL相关，但在SL能量问题中也会出现。
13. Misconceptions and Quick Exam Tips | 常见误区与快速应试技巧
Work is a scalar, not a vector. Do not assign a direction to work.
功是标量，不是矢量。不要给功指明方向。
Kinetic energy depends on speed squared, so doubling speed quadruples kinetic energy.
动能与速度的平方成正比，速度加倍时动能变为原来的四倍。
In energy conservation problems, define the zero potential energy level before solving.
在能量守恒问题中，先定义零势能参考面再解题。
For cyclic processes like a ball returning to its starting height, the net work done by gravity over the complete cycle is zero.
对于小球回到起始高度的循环过程，重力在整个循环中所做的总功为零。
confuse W = Fs with P = Fv. The first requires displacement; the second requires velocity.
不要混淆W = Fs和P = Fv。前者需要位移，后者需要速度。
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