📚 IB Physics: Work, Energy and Power Explained | IB 物理：功、能量与功率精讲
In IB Physics, work, energy and power are linking concepts that explain how forces affect the motion of objects and how energy is transferred from one store to another. Work is the amount of energy transferred when a force acts over a distance, energy is the capacity to do work, and power is the rate at which work is done or energy is transferred.
在 IB 物理中，功、能量与功率是相互联系的概念，它们解释力如何影响物体的运动，以及能量如何从一个储存形式转移到另一个储存形式。功是力在物体位移上作用下所转移的能量，能量是做功的能力，功率则是做功或能量转移的快慢。
1. Work, Energy and Power: The Big Picture | 功、能量与功率：总览
The idea of work in physics is more precise than its everyday meaning. No work is done unless a force causes a displacement. If a person holds a heavy box without moving it, they may feel tired, but in the physics sense, no work is done on the box because the displacement is zero.
物理中对功的定义比日常用语更精确。只有在力的作用下发生位移，才算是做功。如果一个人举着很重的箱子但没有移动，虽然他会感到疲劳，但从物理意义上看，他对箱子并没有做功，因为位移为零。
Energy is the property that makes change possible. It appears in many forms: kinetic energy, gravitational potential energy, elastic potential energy, thermal energy, chemical energy and nuclear energy. Work is a mechanism by which energy is transferred from one body or system to another.
能量是使变化能够发生的属性。它有多种形式：动能、重力势能、弹性势能、热能、化学能和核能。功是能量从一个物体或系统转移到另一个物体或系统的一种方式。
Power tells us how quickly this energy transfer occurs. A more powerful engine does the same amount of work in a shorter time, or more work in the same time.
功率表示能量转移的快慢。功率更大的发动机能在更短时间内完成同样的功，或在相同时间内做更多的功。
2. Work Done by a Constant Force | 恒力做功
For a constant force F applied to an object that moves through a displacement s, the work W done is the product of the component of the force along the displacement and the magnitude of the displacement.
当恒力 F 作用在物体上，并使物体发生位移 s 时，所做的功 W 等于力在位移方向上的分量与位移大小的乘积。
W = F s cos θ
Here θ is the angle between the force vector and the displacement vector. The unit of work is the joule (J), where 1 J = 1 N·m.
其中 θ 是力的方向与位移方向之间的夹角。功的单位是焦耳（J），1 J = 1 N·m。
When the force and displacement are in the same direction, θ = 0° and cos θ = 1, so W = F s. When they are perpendicular, θ = 90° and W = 0. This explains why the normal reaction force does no work when an object slides along a horizontal surface, and why centripetal force does no work on an object moving in a circle.
当力与位移方向相同时，θ = 0°，cos θ = 1，因此 W = F s。当方向垂直时，θ = 90°，W = 0。这解释了为什么物体在水平面上滑动时，支持力不做功；也解释了为什么向心力对做圆周运动的物体不做功。
3. Zero Work and Negative Work | 不做功与负功
Work can be positive, negative or zero. Positive work occurs when the force has a component in the direction of motion, increasing the kinetic energy of the object. Negative work occurs when the force has a component opposite to the motion, decreasing the kinetic energy.
功可能为正、为负或为零。当力的分量与运动方向相同时，做正功，物体动能增加。当力的分量与运动方向相反时，做负功，物体动能减少。
For example, when braking, the frictional force acts opposite to the displacement, so the work done by friction is negative. This is often described as energy being removed from the kinetic energy store of the object.
例如，刹车时摩擦力方向与位移方向相反，因此摩擦力做负功。这通常被描述为能量从物体动能储存中减少。
Zero work is done when the force is perpendicular to the displacement, when the object is stationary, or when the point of application of the force does not move. In IB problems, always identify the force doing the work and carefully measure the angle θ before substituting into the formula.
当力与位移垂直、物体静止或力的作用点不移动时，做功为零。在 IB 习题中，一定要明确是哪个力在做功，并在代入公式前仔细确定角度 θ。
4. Work Done by a Variable Force | 变力做功
A constant force is a special case. In many real situations, the force changes with position, such as a spring force or the force needed to stretch a rubber band. The work done by a variable force can be found graphically.
恒力是一种特殊情况。在许多真实情景中，力随位置变化，例如弹簧的弹力或拉伸橡皮筋所需的力。变力所做的功可以通过图像求解。
The area under a graph of force against displacement gives the work done. This works for constant forces as well, since the area under a horizontal line is a rectangle.
力-位移图像下方的面积等于所做的功。这对恒力同样适用，因为水平线下方的面积是矩形。
W = ∫ F(s) ds
For a non-linear force, the area can be estimated by counting squares on graph paper or by dividing the region into small strips. In the IB syllabus, interpreting such graphs is an important skill.
对于非线性变化的力，可以通过在方格纸上数格子或将区域分成小条来估算面积。在 IB 课程中，理解这种图像是一项重要技能。
5. Kinetic Energy and the Work–Energy Theorem | 动能与动能定理
Kinetic energy is the energy an object has because of its motion. For a mass m moving at speed v, the kinetic energy is given by:
动能是物体由于运动而具有的能量。对于质量为 m、速度为 v 的物体，动能为：
Eₖ = ½ m v²
Because kinetic energy depends on v², doubling the speed quadruples the kinetic energy. This has significant consequences for road safety and collision analysis.
因为动能取决于 v²，速度加倍会使动能变为原来的四倍。这对道路安全和碰撞分析有重要影响。
The work–energy theorem states that the net work done on an object equals its change in kinetic energy:
动能定理指出，对物体所做的净功等于物体动能的变化量：
W_net = ΔEₖ = ½ m v² − ½ m u²
Net work means the work done by the resultant force. If the net work is positive, kinetic energy increases; if negative, kinetic energy decreases.
净功是指合力所做的功。如果净功为正，动能增加；如果净功为负，动能减少。
6. Gravitational Potential Energy | 重力势能
Near the surface of the Earth, the gravitational potential energy of a mass m at a height h above a chosen reference level is given by:
在地球表面附近，质量为 m 的物体在相对所选参考平面高度为 h 处，其重力势能为：
Eₚ = m g h
The reference level is arbitrary; what matters in calculations is the change in height, Δh, because only differences in potential energy are physically meaningful.
参考平面的选择是任意的；在计算中重要的是高度变化 Δh，因为只有势能的变化才具有物理意义。
When an object moves upward, the work done by gravity is negative and gravitational potential energy increases. When an object falls downward, the work done by gravity is positive and gravitational potential energy decreases, usually converting into kinetic energy.
当物体向上运动时，重力做负功，重力势能增加。当物体向下运动时，重力做正功，重力势能减少，通常转化为动能。
For a mass of 2 kg falling through 5 m near the Earth’s surface, the gravitational potential energy released is 2 × 9.8 × 5 = 98 J. This energy transfers to kinetic energy if air resistance is negligible.
一个质量为 2 kg 的物体在地球表面附近下落 5 m，释放的重力势能为 2 × 9.8 × 5 = 98 J。若空气阻力可忽略，这些能量将转化为动能。
7. Elastic Potential Energy | 弹性势能
An ideal spring obeys Hooke’s law: the restoring force is proportional to the extension or compression x, with spring constant k.
理想弹簧遵循胡克定律：弹力与伸长量或压缩量 x 成正比，比例系数为劲度系数 k。
F = k x
Because the force changes linearly with extension, the work done in stretching the spring is the area under the F–x graph, which is a triangle of base x and height kx.
由于力随伸长量线性变化，拉伸弹簧所做的功等于 F–x 图像下方的面积，即底为 x、高为 kx 的三角形面积。
Eₑ = ½ k x²
This elastic potential energy is stored in the spring and can be released later, for example in a catapult, a clock spring, or a bouncing ball. Also note that the work done to stretch a spring depends on x², so compressing a spring twice as far requires four times the energy.
这种弹性势能储存在弹簧中，并可在之后释放，例如弹弓、钟表发条或弹跳球中。注意，拉伸弹簧所需的功与 x² 有关，因此将弹簧压缩两倍距离需要四倍的能量。
8. Power | 功率
Power is the rate at which work is done or energy is transferred. The average power is:
功率是做功或能量转移的速率。平均功率为：
P = W / t
The unit of power is the watt (W), where 1 W = 1 J/s. IB problems often ask you to convert between kilowatts, megawatts and joules, so always check units carefully.
功率的单位是瓦特（W），1 W = 1 J/s。IB 题目经常要求换算千瓦、兆瓦和焦耳，因此务必仔细检查单位。
For a force F acting on an object moving at speed v, the instantaneous power can be related to force and velocity:
当力 F 作用在速度为 v 的物体上时，瞬时功率可以与力和速度联系起来：
P = F v cos θ
If the force is in the direction of motion, then P = F v. This is useful for calculating the power needed by vehicles climbing hills, accelerating, or overcoming air resistance.
如果力的方向与运动方向相同，则 P = F v。这在计算车辆爬坡、加速或克服空气阻力所需的功率时非常有用。
9. Efficiency | 效率
No machine transfers all input energy into useful output energy. Some energy is always lost to the surroundings, often as thermal energy due to friction and air resistance. Efficiency compares the useful output with the total input.
没有任何机器能将输入的能量全部转化为有用的输出能量。总有一部分能量会散失到周围环境中，通常是由摩擦和空气阻力导致的热能。效率用于比较有用输出与总输入。
Efficiency = (Useful output energy / Total input energy) × 100%
Efficiency can also be expressed in terms of power:
效率也可以用功率来表示：
Efficiency = (Useful output power / Total input power) × 100%
For example, an electric motor that receives 500 W of electrical power and produces 350 W of mechanical power has an efficiency of 70%. The remaining 150 W is transferred to thermal energy in the motor and surroundings.
例如，一个电动机输入电功率为 500 W，输出机械功率为 350 W，其效率为 70%。其余 150 W 转化为电动机和周围环境的热能。
10. Conservation of Mechanical Energy | 机械能守恒
The law of conservation of energy states that energy cannot be created or destroyed, only transferred or converted from one form to another. In a system where only conservative forces act, such as gravity or an ideal spring, mechanical energy is conserved.
能量守恒定律指出，能量不能被创造或消灭，只能从一种形式转移到另一种形式或相互转化。在只有保守力（如重力或理想弹簧）作用的系统中，机械能守恒。
Eₖ + Eₚ = constant
For a pendulum swinging with negligible air resistance, gravitational potential energy at the highest point is converted into kinetic energy at the lowest point, and then back again. The total mechanical energy remains constant throughout.
对于空气阻力可忽略的摆动摆锤，最高点的重力势能转化为最低点的动能，然后再转化回去。整个过程中总机械能保持不变。
When non-conservative forces such as friction or air resistance do work, mechanical energy is not conserved. Some mechanical energy is transformed into thermal energy, so the final total mechanical energy is less than the initial value.
当摩擦力或空气阻力等非保守力做功时，机械能不守恒。部分机械能转化为热能，因此最终的机械能总量小于初始值。
11. Exam Tips and Common Mistakes | 考点提示与常见错误
IB exam questions often combine work, energy and power with motion, forces and graphs. One common mistake is using the full force instead of the component in the direction of displacement. Always check the angle θ.
IB 考题常将功、能量和功率与运动、力的知识以及图像分析结合。常见错误是使用整个力而不是沿位移方向的分量。一定要检查角度 θ。
Another frequent error is confusing mass and weight. Gravitational potential energy uses mass m, not the force of gravity. In an exam question, the value 9.8 N/kg can be used either as gravitational field strength or as acceleration due to gravity.
另一个常见错误是混淆质量与重力。重力势能使用的是质量 m，而不是重力的大小。在考题中，9.8 N/kg 既可以用作重力场强度，也可以用作重力加速度。
Students also mix up energy and power. Energy is measured in joules and power is measured in watts; a watt is a joule per second. A question might ask for the work done in joules, not the power in watts.
学生也经常混淆能量和功率。能量单位是焦耳，功率单位是瓦特；1 瓦特等于 1 焦耳每秒。题目可能要求计算以焦耳为单位的功，而不是以瓦特为单位的功率。
When using the work–energy theorem, remember that W_net includes the total work done by all forces. If a problem asks for the work done by one particular force, use a free-body diagram to identify each force and its displacement.
使用动能定理时，记住 W_net 是合力所做的总功。如果题目要求某个力做的功，应先画受力分析图，明确每个力及其位移。
Graph questions may require you to find the gradient or the area. In a force–displacement graph, the area is work. In a power–time graph, the area is energy. In a work–time graph, the gradient is power. Read the axes before applying a formula.
图像题可能要求你求斜率或面积。在力-位移图像中，面积表示功。在功率-时间图像中，面积表示能量。在功-时间图像中，斜率表示功率。在套用公式前，先仔细阅读坐标轴。
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