📚 Work and Energy | 功和能量
In IB Mathematics, 'work and energy' bridges physical concepts with core calculus and vector techniques. Whether analysing the scalar product for constant forces or integrating to handle variable forces, this topic demonstrates how mathematical tools model mechanical reality. Understanding these links not only prepares you for applied exam questions but deepens your grasp of integration, differentiation and vector geometry.
在 IB 数学中，“功与能量”将物理概念与核心微积分和向量技巧连接起来。无论是用标量积分析恒力，还是通过积分处理变力，这个主题都展示了数学工具如何模拟力学现实。理解这些联系不仅能帮你在应用题中得分，还能加深你对积分、微分和向量几何的掌握。
1. Work Defined by Scalar Product | 标量积定义的功
Work is a scalar quantity obtained by the dot product of force vector F and displacement vector d: W = F ⋅ d = |F||d| cos θ, where θ is the angle between the force and the direction of motion. Only the component of force along the displacement does work; a perpendicular force does zero work.
功是由力向量 F 与位移向量 d 的点积得到的标量：W = F ⋅ d = |F||d| cos θ，其中 θ 是力与运动方向之间的夹角。只有沿位移方向的分力做功；垂直力做功为零。
The scalar product formulation is central to IB vector problems. You must be comfortable computing F ⋅ d using component forms: if F = (F₁, F₂) and d = (d₁, d₂), then W = F₁d₁ + F₂d₂. This avoids the need to determine θ explicitly in many coordinate-based questions.
标量积公式是 IB 向量问题的核心。你必须熟练使用分量形式计算 F ⋅ d：若 F = (F₁, F₂) 且 d = (d₁, d₂)，则 W = F₁d₁ + F₂d₂。在许多基于坐标的题目中，这可以省去明确求 θ 的步骤。
2. Constant Force and Displacement | 恒力与位移
When a constant force acts on an object moving in a straight line, the work done is simply W = F d cos θ. Typical exam contexts include inclined planes, where the force of gravity is constant, or a constant applied tension. The mathematical challenge often lies in resolving vectors and identifying the correct angle.
当恒力作用在沿直线运动的物体上时，做功就是 W = F d cos θ。典型的考题情境包括斜面（重力恒定）或恒定的拉力。数学上的挑战通常在于分解向量并确定正确的夹角。
Watch out for displacement not parallel to the force. For example, if an object is lifted vertically by a force that acts at an angle, only the vertical component multiplied by the vertical displacement contributes to the work. Many mistakes arise from using the full magnitude of F without considering cos θ.
注意位移不与力平行的情况。例如，若物体被一个成角度的力垂直提起，只有垂直分量乘以垂直位移才计为功。很多错误都源于不加 cos θ 就直接使用 F 的全部大小。
3. Work Done by a Variable Force (1D) | 一维变力做功
For a force that changes with position, work is found by definite integration: W = ∫x₁x₂ F(x) dx. This is a direct application of the area under a force–distance graph. In IB Mathematics, you will see functions like F(x) = kx (spring) or F(x) = 3x² + 2.
对于随位置变化的力，功通过定积分求得：W = ∫x₁x₂ F(x) dx。这是力–距离图像下方面积的直接应用。在 IB 数学中，你会遇到诸如 F(x) = kx（弹簧）或 F(x) = 3x² + 2 之类的函数。
The integration must respect the sign of F(x). If the force opposes the motion (e.g. friction, which would be negative in the chosen positive direction), the integral gives a negative value, reducing the total kinetic energy. Always define your positive direction before setting up the limits.
积分时必须尊重 F(x) 的符号。如果力与运动方向相反（例如摩擦力，在所取正方向下为负），积分会给出负值，从而减少总动能。务必在确定积分限之前先定义正方向。
4. Work–Energy Theorem and Kinetic Energy | 动能定理与动能
The net work done on an object equals its change in kinetic energy: Wnet = ΔKE = ½ m v² − ½ m u². This theorem, derivable from Newton’s second law and integration, provides a powerful shortcut: you can find speed without calculating acceleration or time.
物体所受的净功等于其动能的变化：Wnet = ΔKE = ½ m v² − ½ m u²。该定理可由牛顿第二定律和积分推导出来，提供了一个有力的捷径：无需计算加速度或时间就能求出速度。
∫x₁x₂ Fnet dx = ½ m v₂² − ½ m v₁²
IB exam questions often ask you to use the work–energy principle after computing work via integration. Just remember that Wnet is the sum of work by all forces, including gravity, friction and applied forces.
IB 考试经常在让你通过积分算出功之后，再要求用动能定理。只需记住 Wnet 是所有力做功的代数和，包括重力、摩擦力和外力。
5. Potential Energy from Integration | 积分求势能
Conservative forces allow us to define potential energy U such that the work done by the force is the negative change in U: Wcons = −ΔU. For a one‑dimensional conservative force F(x), we obtain ΔU = −∫x₁x₂ F(x) dx. Choosing a reference point sets the constant of integration.
保守力使我们能定义势能 U，使得保守力做的功等于势能的减少：Wcons = −ΔU。对于一维保守力 F(x)，有 ΔU = −∫x₁x₂ F(x) dx。选取一个参考点就确定了积分常数。
For gravity near Earth’s surface, F = mg (downwards). Integrating with an upward positive axis gives U = mgh if we set U=0 at h=0. For an ideal spring, F = −kx, so U = ½ kx² when the unstretched position is taken as zero.
对于地表附近的重力，F = mg（向下）。取向上为正轴积分，并设 h=0 处 U=0，即得 U = mgh。对于理想弹簧，F = −kx，因此以原长为零点时，U = ½ kx²。
6. Power as the Derivative of Work | 功的导数——功率
Power is the rate at which work is done: P = dW/dt. Using the chain rule and v = dx/dt, we also find P = F·v for a constant force, or P = F(x) v for a position‑dependent force. This blends differentiation with physical meaning.
功率是做功的速率：P = dW/dt。利用链式法则和 v = dx/dt，我们得到恒力下的 P = F·v，或位置依赖力下的 P = F(x)v。这融合了微分与物理意义。
In problems involving variable velocity, you might be given v(t) and F as a function of t; then instantaneous power is just the product F(t)·v(t). Questions sometimes ask for average power over a time interval, which requires integrating P(t) with respect to t and dividing by the time interval.
在涉及变速的问题中，你可能得到 v(t) 和作为 t 的函数的 F；那么瞬时功率就是乘积 F(t)·v(t)。有些题目会要求某段时间内的平均功率，这时需要对 P(t) 积分并除以时间间隔。
7. Conservation of Mechanical Energy | 机械能守恒
When only conservative forces do work, the sum of kinetic and potential energy remains constant: E = KE + U = constant. This principle translates a dynamical problem into an algebraic one: ½ m v₁² + U₁ = ½ m v₂² + U₂. It is especially handy for systems involving gravity or springs.
当只有保守力做功时，动能与势能之和保持不变：E = KE + U = 恒量。这一原理把动力学问题转化为代数问题：½ m v₁² + U₁ = ½ m v₂² + U₂。对于涉及重力或弹簧的系统，它特别方便。
Be cautious: if friction or an external applied force does work, mechanical energy is not conserved. You must then use the more general work–energy theorem, writing Wother = ΔE, where E is total mechanical energy.
注意：如果摩擦力或外力做功，机械能不守恒。此时必须使用更普遍的动能定理，写作 W其他 = ΔE，其中 E 是总机械能。
8. Typical Exam Problems and Strategies | 经典考题与策略
IB exam questions often combine several concepts: for instance, you might integrate F(x) to find work, then apply the work–energy theorem to find speed, and finally compute power at an instant. The following table summarises common problem types and key steps.
IB 考题常将多个概念结合起来：例如，你可能要对 F(x) 积分求功，然后应用动能定理求速度，最后计算瞬时功率。下表总结了常见题型和关键步骤。
| Problem type 题型 | Mathematical tool 数学工具 | Key caution 注意事项 |
|---|---|---|
| Lifting an object at constant speed | W = mgh, dot product | Use the vertical displacement, not path length |
| Spring launch / compression | ∫ F dx, ½ kx² energy conservation | Set the unstretched position as x=0 carefully |
| Variable force from a graph | Area under F–x graph (integration/counting squares) | Negative area means negative work |
| Power output of a motor | P = Fv or dW/dt | Distinguish average from instantaneous power |
| Inclined plane with friction | Wnet = ΔKE, Wfric = −μ N d | Include all forces in net work |
A structured problem‑solving approach helps: (1) draw and label forces, (2) choose a coordinate system, (3) write expressions for work, (4) integrate if needed, (5) apply energy principles. With practice, these problems become predictable and highly scoring.
一个结构化的解题方法大有裨益：(1) 画图并标出各个力，(2) 选取坐标系，(3) 写出功的表达式，(4) 必要时代入积分，(5) 应用能量原理。通过练习，这类题目会变得容易预测且得分很高。
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