📚 Formula Derivation in Activate Physics Student Book | Activate Physics 公式推导
In Activate Physics, students encounter many key formulas that describe how the physical world works. Understanding the derivation of these formulas, not just memorising them, helps build a solid foundation in physics. This article walks through the logical steps behind some of the fundamental equations found in the Activate Physics Student Book, from speed to Ohm’s law.
在《Activate Physics》中，学生会遇到许多描述物理世界运作方式的关键公式。理解这些公式的推导过程，而不仅仅是记忆它们，有助于打下扎实的物理基础。本文将一步步讲解《Activate Physics 学生用书》中一些基本方程背后的逻辑步骤，从速度公式到欧姆定律。
1. Speed Formula (v = s/t) | 速度公式 (v = s/t)
Speed is a measure of how quickly an object covers distance. It is defined as the distance travelled divided by the time taken. Mathematically, this is expressed as:
速度是衡量物体移动快慢的量。它定义为单位时间内通过的距离。数学表达式为：
v = s / t
where v represents speed, s stands for distance and t is the time taken. The derivation is straightforward because it comes directly from the definition. For example, if a cyclist rides 30 metres in 6 seconds, the average speed is 30 ÷ 6 = 5 m/s.
其中 v 代表速度，s 代表距离，t 代表所用时间。由于此公式直接源于定义，推导过程非常直接。例如，一位骑自行车的人 6 秒内行驶了 30 米，平均速度就是 30 ÷ 6 = 5 米/秒。
We can also rearrange the formula to calculate distance or time when the other two quantities are known: s = v t and t = s / v. These variations are frequently used in problem-solving.
我们还可以重新排列公式，在已知另外两个量时求解距离或时间：s = v t 以及 t = s / v。这些变形在解题中经常用到。
2. Density Formula (ρ = m/V) | 密度公式 (ρ = m/V)
Density describes how much mass is packed into a given volume. It is defined as mass per unit volume. The formula is therefore:
密度描述了单位体积内含有多少质量。它定义为单位体积的质量。因此公式为：
ρ = m / V
where the Greek letter ρ (rho) represents density, m is mass and V is volume. This formula can be derived by measuring the mass of a substance and the space it occupies. If a block of metal has a mass of 200 g and a volume of 50 cm³, its density is 200 ÷ 50 = 4 g/cm³.
希腊字母 ρ 表示密度，m 表示质量，V 表示体积。通过测量物质的质量和它所占据的空间，就能推导出这一公式。如果一块金属的质量为 200 g，体积为 50 cm³，它的密度就是 200 ÷ 50 = 4 g/cm³。
Rearranging gives m = ρ V and V = m / ρ, which are useful when predicting the mass of a known volume or determining the volume of a certain mass of material.
重新排列后得到 m = ρ V 和 V = m / ρ，这在已知体积预测质量或已知质量求体积时非常实用。
3. Pressure Formula (p = F/A) | 压强公式 (p = F/A)
Pressure is the effect of a force spread over an area. It is defined as the force acting perpendicularly per unit area. The formula is:
压强是力分布在一个面积上产生的效果。它定义为单位面积上垂直作用的力。公式为：
p = F / A
where p is pressure, F is the force and A is the area over which the force is applied. The derivation comes from considering how the same force produces a larger pressure on a smaller area. A woman wearing stiletto heels exerts a greater pressure on the floor than an elephant’s foot because her weight is concentrated on a very small area.
其中 p 为压强，F 为作用力，A 为受力面积。该公式的推导基于这一原理：相同的力作用在越小的面积上，产生的压强越大。一位穿细高跟鞋的女性对地面施加的压强可能比大象的脚还大，因为她的体重集中在极小的面积上。
If a force of 500 N acts on an area of 0.25 m², the pressure is 500 ÷ 0.25 = 2000 Pa. This relationship is used in designing snow shoes or tank tracks to reduce pressure by increasing area.
如果 500 N 的力作用在 0.25 m² 的面积上，压强为 500 ÷ 0.25 = 2000 Pa。设计雪鞋或坦克履带就是通过增大面积来减小压强，利用了这一关系。
4. Weight Formula (W = mg) | 重力公式 (W = mg)
Weight is the gravitational force acting on a mass. Newton’s second law states that force equals mass times acceleration (F = ma). Near the Earth’s surface, the acceleration due to gravity is denoted by g, so the weight force W is:
重量是作用在质量上的引力。牛顿第二定律指出力等于质量乘以加速度 (F = ma)。在地球表面附近，重力加速度记为 g，因此重量 W 为：
W = m g
where g is the gravitational field strength, approximately 9.8 m/s² on Earth. This derivation shows that weight and mass are proportional; an object’s weight changes on different planets because g varies, but its mass remains constant.
其中 g 为重力场强度，地球上约为 9.8 m/s²。这个推导表明重量与质量成正比；物体的重量在不同行星上会发生变化，因为 g 值不同，但其质量保持不变。
For example, a 2 kg mass on Earth weighs W = 2 × 9.8 = 19.6 N. On the Moon, where g ≈ 1.6 m/s², the same mass would weigh only 3.2 N.
例如，地球上 2 kg 的物体重量 W = 2 × 9.8 = 19.6 N。在月球上 g ≈ 1.6 m/s²，同样的质量重量仅为 3.2 N。
5. Kinetic Energy Formula (Ek = ½mv²) | 动能公式 (Ek = ½mv²)
Kinetic energy is the energy an object possesses due to its motion. To derive the formula, consider a constant force F acting on a mass m initially at rest, moving it through a distance s. The work done on the object is W = F s. Using F = ma and one of the equations of motion: v² = u² + 2 a s, with initial velocity u = 0, we get v² = 2 a s.
动能是物体因运动而具有的能量。为推导该公式，考虑一个恒力 F 作用在原先静止的质量 m 上，使其移动距离 s。对物体所做的功为 W = F s。利用 F = ma 和运动学公式 v² = u² + 2 a s，初速度 u = 0，得到 v² = 2 a s。
Eₖ = ½ m v²
Substituting a s from v² = 2 a s gives a s = v²/2. Then the work done W = m a s = m × (v²/2) = ½ m v². This work is entirely transferred to kinetic energy, so Ek = ½ m v².
由 v² = 2 a s 可得 a s = v²/2。代入做工表达式得 W = m a s = m × (v²/2) = ½ m v²。这部分功完全转化为动能，因此 Ek = ½ m v²。
This formula tells us that kinetic energy depends on the square of the speed. Doubling the speed quadruples the kinetic energy, which is why high-speed collisions are so devastating.
该公式表明动能取决于速度的平方。速度加倍，动能变为四倍，这就是高速碰撞破坏力巨大的原因。
6. Gravitational Potential Energy Formula (Ep = mgh) | 重力势能公式 (Ep = mgh)
Gravitational potential energy is the energy stored in an object due to its height above the ground. To lift an object through a vertical height h, a force equal to its weight mg must be applied upwards. The work done against gravity is:
重力势能是物体因其离地高度而储存的能量。要将物体竖直提升高度 h，必须向上施加一个等于其重量 mg 的力。克服重力所做的功为：
Ep = m g h
Work done = force × distance moved in direction of force = (mg) × h = mgh. Since the work done is stored as gravitational potential energy, we have Ep = mgh. This assumes g is constant near the Earth’s surface.
做功 = 力 × 沿力方向移动的距离 = (mg) × h = mgh。由于所做的功以重力势能的形式储存起来，因此 Ep = mgh。这里假设在地表附近 g 为常数。
If a 5 kg box is lifted onto a shelf 2 m high, the gain in Ep is 5 × 9.8 × 2 = 98 J. The same energy is released if the box falls.
若将 5 kg 的箱子提到 2 m 高的架子上，Ep 增加 5 × 9.8 × 2 = 98 J。箱子掉落时则会释放相同的能量。
7. Work Done Formula (W = F d) | 做功公式 (W = F d)
Work is done whenever a force moves an object in the direction of the force. The amount of work is defined as the product of the force and the distance moved. This gives the simple formula:
当一个力使物体沿力的方向移动时，就做了功。功的大小定义为力与移动距离的乘积。由此得到简单公式：
W = F d
where W is work in joules (J), F is the constant force in newtons (N), and d is the distance moved in metres (m). If the force is not parallel to the motion, only the component along the motion does work. In many Activate Physics problems, forces act in a straight line, so this form is sufficient.
其中 W 为功，单位焦耳 (J)；F 为恒力，单位牛顿 (N)；d 为移动距离，单位米 (m)。如果力与运动方向不平行，只有沿运动方向的分力做功。在《Activate Physics》的许多问题中，力沿直线作用，这一形式已经够用。
For instance, pushing a box with a force of 20 N over a distance of 3 m does 20 × 3 = 60 J of work. No work is done if the object does not move, no matter how large the force.
例如，用 20 N 的力推一个箱子移动 3 m，做功 20 × 3 = 60 J。如果物体没有移动，无论力多大都不做功。
8. Ohm’s Law (V = I R) | 欧姆定律 (V = I R)
Ohm’s law relates the voltage across a conductor to the current through it and its resistance. It was derived experimentally by Georg Ohm, who found that for many materials, the current is directly proportional to the voltage when temperature is constant. The constant of proportionality is the resistance R:
欧姆定律将导体两端的电压与通过导体的电流及其电阻联系起来。它由乔治·欧姆通过实验推导得出：对于许多材料，温度恒定时电流与电压成正比。比例常数就是电阻 R：
V = I R
where V is the potential difference in volts, I is the current in amperes, and R is the resistance in ohms (Ω). This formula can be rearranged to I = V / R and R = V / I, which are all forms used in circuit analysis.
其中 V 是电势差，单位伏特；I 是电流，单位安培；R 是电阻，单位欧姆 (Ω)。该公式可变形为 I = V / R 和 R = V / I，这些都是电路分析中常用的形式。
An example of its use: if a lamp with a resistance of 10 Ω has a current of 0.5 A flowing through it, the voltage across it must be V = 0.5 × 10 = 5 V. Ohm’s law holds for ohmic conductors, but not for all components like diodes.
应用举例：若一盏电阻为 10 Ω 的灯通过 0.5 A 的电流，其两端电压必为 V = 0.5 × 10 = 5 V。欧姆定律适用于欧姆导体，但不适用于二极管等所有元件。
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