Year 12 SQA Physics: Formula & Theorem Quick Reference Handbook | SQA 物理公式定理速查手册

📚 Year 12 SQA Physics: Formula & Theorem Quick Reference Handbook | SQA 物理公式定理速查手册

Welcome to the Year 12 SQA Physics Formula & Theorem Quick Reference Handbook. This comprehensive guide brings together all the essential formulas, theorems, and key relationships you are expected to know for the SQA Higher Physics course. The content is organised into the two major units – Our Dynamic Universe and Electricity, Particles and Waves – and is designed for rapid revision and deep understanding. Use it regularly to build confidence, reinforce connections, and perform excellently in your assessments and exams.

欢迎使用 SQA 物理公式定理速查手册。本手册汇集了 SQA 高等物理课程（Higher Physics）所需的所有核心公式、定理和重要关系，涵盖“我们的动态宇宙”和“电学、粒子与波”两大单元。手册专为快速复习和深入理解而设计，助你巩固知识、建立联系，在考核与考试中发挥出色。

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1. Uncertainties and Errors | 不确定度与误差

Absolute uncertainty (Δx) represents the range within which the true value of a measurement is expected to lie. It can be determined by the precision of the instrument or by repeating measurements and finding the half-range.

绝对不确定度 (Δx) 表示测量真值可能落入的范围。它可通过仪器精度或重复测量求半范围来确定。

Fractional uncertainty = Δx / x, Percentage uncertainty = (Δx / x) × 100%

Fractional and percentage uncertainties are dimensionless measures that allow comparison of the size of the uncertainty relative to the measurement.

相对不确定度和百分不确定度是无量纲量，便于比较不确定度相对于测量值的大小。

When two quantities are added or subtracted, the absolute uncertainties add directly: if z = x + y or z = x − y, then the uncertainty in z is given by Δz = Δx + Δy.

当两个量相加或相减时，绝对不确定度直接相加：若 z = x + y 或 z = x − y，则 z 的不确定度 Δz = Δx + Δy。

z = xy → Δz/z = Δx/x + Δy/y
z = x / y → Δz/z = Δx/x + Δy/y

For multiplication and division, we add fractional (or percentage) uncertainties. If a quantity is raised to a power, for example z = xⁿ, the fractional uncertainty is multiplied by the absolute value of the power: Δz/z = |n| Δx/x.

对于乘法和除法，我们将相对不确定度（或百分不确定度）相加。若量被乘方，如 z = xⁿ，则相对不确定度乘以幂的绝对值：Δz/z = |n| Δx/x。

The mean of a set of readings is given by x̄ = Σx / N, and the random uncertainty in the mean can be estimated using the standard deviation of the sample or by the range method.

一组读数的平均值 x̄ = Σx / N，其平均值的随机不确定度可用样本标准差或极差法估算。

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2. Kinematics | 运动学

The three equations of motion apply to objects moving with constant acceleration in a straight line. They link initial velocity u, final velocity v, acceleration a, displacement s and time t.

三个运动学方程适用于匀加速直线运动，联系了初速度 u、末速度 v、加速度 a、位移 s 和时间 t。

v = u + at
s = ut + ½at²
v² = u² + 2as

An additional useful relation is s = ½(u + v)t, which gives displacement when average velocity is multiplied by time.

另一个常用关系为 s = ½(u + v)t，即平均速度与时间的乘积。

On a velocity–time graph, the gradient at any point equals the acceleration, and the area under the graph equals the displacement. For a projectile, the horizontal and vertical motions are independent. The horizontal component of velocity remains constant (there is no horizontal acceleration), while the vertical component changes due to gravity.

在速度–时间图上，任意点的斜率等于加速度，图线下面积等于位移。对于抛体，水平和竖直运动彼此独立。水平分速度保持不变（无水平加速度），竖直分速度因重力而改变。

Horizontal: vₓ = u cosθ, sₓ = u cosθ × t
Vertical: vᵧ = u sinθ − gt, sᵧ = u sinθ × t − ½gt²

The maximum height reached by a projectile is given by (u sinθ)² / (2g), and the total time of flight is (2u sinθ)/g, assuming the launch and landing are at the same vertical level. The range is R = (u² sin 2θ)/g.

抛体达到的最大高度为 (u sinθ)² / (2g)，飞行总时间为 (2u sinθ)/g，假定起落点在同一水平面。水平射程为 R = (u² sin 2θ)/g。

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3. Dynamics and Forces | 动力学与力

Newton’s second law states that the net force acting on an object equals its mass multiplied by its acceleration: F = ma. In problems you must first identify all forces and find the resultant (unbalanced) force in the direction of motion.

牛顿第二定律指出，作用在物体上的合力等于质量乘以加速度：F = ma。解题时必须先识别所有力，求出运动方向上的合外力。

The weight of an object near the Earth’s surface is W = mg, where g ≈ 9.8 m s⁻². The normal reaction force is perpendicular to the contact surface. Tension is the pulling force transmitted through a string or cable.

地球表面附近物体的重量为 W = mg，g ≈ 9.8 m s⁻²。法向反作用力垂直于接触面，张力是通过绳或缆传递的拉力。

When forces act at an angle, they can be resolved into perpendicular components: for a force F at an angle θ to the horizontal, Fₓ = F cosθ and Fᵧ = F sinθ.

当力成一定角度时，可分解为垂直分量：若力 F 与水平夹角 θ，则 Fₓ = F cosθ，Fᵧ = F sinθ。

Friction is described by two cases. The maximum static friction is F_f ≤ μₛR, where μₛ is the coefficient of static friction and R is the normal reaction. Kinetic (sliding) friction is given by F_f = μₖR.

摩擦力分为两种情况。最大静摩擦力 F_f ≤ μₛR，μₛ 为静摩擦系数，R 为法向反力。滑动摩擦力为 F_f = μₖR。

An object falling through a fluid reaches terminal velocity when the upward drag force plus upthrust equals the weight; at that stage the net force is zero and acceleration ceases.

物体在流体中下落时，当向上的阻力加浮力等于重力时，达到终极速度；此时合力为零，加速度消失。

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4. Momentum and Impulse | 动量与冲量

Linear momentum is defined as the product of mass and velocity: p = mv. Momentum is a vector quantity and its direction is the same as the velocity.

线动量定义为质量与速度的乘积：p = mv。动量是矢量，方向与速度相同。

Impulse is the change in momentum and is equal to the product of the average resultant force and the time interval over which it acts: Δp = F Δt. On a force–time graph, the area under the curve represents the impulse.

冲量等于动量的变化，也等于平均合力与作用时间的乘积：Δp = F Δt。在力–时间图上，曲线下方面积表示冲量。

The law of conservation of momentum states that in the absence of external forces, the total momentum of an isolated system remains constant. For a collision or explosion: Σp_before = Σp_after.

动量守恒定律指出，在无外力作用下，孤立系统的总动量保持不变。对于碰撞或爆炸：系统总动量（前）= 系统总动量（后）。

In elastic collisions, both momentum and kinetic energy are conserved. In inelastic collisions, momentum is conserved but kinetic energy is not – some energy is transformed into heat, sound or deformation.

在弹性碰撞中，动量和动能都守恒。在非弹性碰撞中，动量守恒，但动能不守恒——部分能量转变为热、声或形变。

Explosion: m₁v₁ + m₂v₂ = 0 (initially at rest)

For explosions where the system is initially stationary, total momentum remains zero, and the fragments move in opposite directions with momenta equal in magnitude.

对于初始静止的系统发生爆炸，总动量依然为零，碎片以等大反向的动量分开。

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5. Work, Energy and Power | 功、能与功率

Work is done when a force moves an object through a displacement in the direction of the force. It is calculated as W = F d cosθ, where θ is the angle between the force and the displacement. Work is measured in joules (J).

功是力在位移方向上作用的结果，计算公式为 W = F d cosθ，其中 θ 为力与位移的夹角。功的单位是焦耳（J）。

Kinetic energy is the energy an object possesses due to its motion: Eₖ = ½mv². Gravitational potential energy depends on height: Eₚ = mgh.

动能是物体因运动而具有的能量：Eₖ = ½mv²。重力势能取决于高度：Eₚ = mgh。

Eₖ + Eₚ = constant (if only conservative forces act)

When only gravity (or other conservative forces) does work, the total mechanical energy of the system is conserved. Real systems lose some mechanical energy to friction, which appears as internal (heat) energy.

当只有重力（或其他保守力）做功时，系统总机械能守恒。真实系统因摩擦会损失部分机械能，转化为内能（热能）。

Power is the rate of doing work or transferring energy: P = W / t = ΔE / t. For an object moving at constant speed v under force F, power can be expressed as P = Fv.

功率是做功或能量转换的速率：P = W / t = ΔE / t。当物体在力 F 作用下以恒定速度 v 运动时，功率可表示为 P = Fv。

Efficiency is the ratio of useful output energy (or power) to total input energy, usually expressed as a percentage: efficiency = (E_out / E_in) × 100%.

效率是有用输出能量（或功率）与总输入能量之比，通常用百分数表示：效率 = (E_out / E_in) × 100%。

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6. Circular Motion | 圆周运动

An object moving in a circular path with constant speed undergoes uniform circular motion. The angular velocity ω (rad s⁻¹) is defined as the rate of change of angle: ω = θ / t. It is related to the period T (time for one complete revolution) and frequency f by ω = 2π / T = 2πf.

物体以恒定速率做圆周运动称为匀速圆周运动。角速度 ω（rad s⁻¹）定义为角度变化率：ω = θ / t。它与周期 T（转过一圈的时间）和频率 f 的关系为 ω = 2π / T = 2πf。

Linear speed: v = rω

The linear speed v along the tangent is related to the radius r and angular velocity ω. The direction of the velocity is constantly changing, which implies an acceleration towards the centre.

线速度 v 沿切线方向，与半径 r 和角速度 ω 相关：v = rω。速度的方向不断改变，因此存在指向圆心的加速度。

Centripetal acceleration: a = v² / r = rω²

Centripetal

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