📚 AS Physics Unit 2 Mark Scheme Jan22: Formula Derivations | AS物理单元2评分标准2022年1月：公式推导
The January 2022 AS Physics Unit 2 mark scheme places significant emphasis on students' ability to derive key equations from first principles. This article revisits those formula derivations, unpacking the logic and assumptions behind each one. Whether it is kinematics, materials, waves, or electricity, understanding the derivations not only helps you secure marks but also deepens your grasp of the underlying physics.
2022年1月AS物理单元2的评分标准特别看重学生从基本原理推导核心公式的能力。本文重新梳理这些公式的推导过程，剖析每一步的逻辑和假设。无论是运动学、材料、波还是电学，真正理解推导不仅能帮你稳拿分数，还能加深你对物理本质的理解。
1. Deriving the SUVAT Equations | 推导匀加速运动方程
nge of velocity. For constant acceleration a, we have a = (v - u)/t, which directly rearranges to v = u + at. From this, the area under a velocity– e velocity multiplied by time: s = ½(u + v)t. Substituting v = u + at into this average yields s = ut + ½at². Another common approach is to eliminate t by rearranging v = u + at into t = (v - u)/a and substituting into s = ½(u + v)t, giving v² = u² + 2as.
SUVAT方程是AS运动学的基石。评分标准要求从加速度的定义——速度变化率——出发。对于恒定加速度a，有a = (v - u)/t，直接整理得v = u + at。由此，利用速度-时间图像的面积可以求位移。画出v-t图，图形是一个梯形（或矩形加三角形）。位移s等于平均速度乘以时间：s = ½(u + v)t。将v = u + at代入平均速度即得s = ut + ½at²。另一种常见推导是消去t：由v = u + at解出t = (v - u)/a，代入s = ½(u + v)t，得到v² = u² + 2as。
The key assumptions are constant acceleration and motion along a straight line. These derivations appear frequently in past papers, and the January 2022 mark scheme awarded marks for clearly stating the initial definition and showing algebraic manipulation step by step.
关键假设是加速度恒定且沿直线运动。这些推导在历年考题中反复出现，2022年1月评分标准对清晰陈述初始定义、逐步展示代数变换给予分数。
2. Work–Energy Principle and Kinetic Energy | 功–能原理与动能
ion of work done by a constant force: W = F s. If the force causes an acceleration, Newton' ²)/(2) = ½mv² - ½mu². The mark scheme often requires you to link the net work done to the change in kinetic energy. Notably, this derivation assumes the force is constant and acts in the direction of motion.
功与动能之间的关系是一个经典推导。从恒力做功的定义出发：W = F s。若此力产生加速度，由牛顿第二定律F = m a。利用运动学方程v² = u² + 2as，可将a s表示为(v² - u²)/2。代入功的表达式：W = m × (v² - u²)/2 = ½mv² - ½mu²。评分标准常要求你将合力做功与动能变化联系起来。注意，这一推导假设力是恒定的且与运动方向相同。
In the Jan22 scheme, candidates were rewarded for recognising that if the only force doing work is the net force, then the work done equals the change in kinetic energy. Examiners also looked for correct handling of negative work when forces oppose motion.
在2022年1月评分标准中，学生若认识到若做功的力仅含合力，则做功等于动能变化量，即可得分。考官还关注当力阻碍运动时，负功处理是否正确。
3. Conservation of Momentum from Newton's Third Law | 从牛顿第三定律推导弹道动量守恒
Newton's third law states that when two bodies interact, the force exerted by A on B is equal in magnitude and opposite in direction to the force exerted by B on A. If the interaction lasts for a time Δt, then the impulse on B is FAB Δt and the impulse on A is FBA Δt. Since FAB = -FBA, the impulses are equal and opposite. Impulse equals change in momentum, so ΔpB = -ΔpA, leading to ΔpA + ΔpB Thus, the total momentum of the system remains constant in the absence of external forces.
牛顿第三定律指出，两物体相互作用时，A对B的力与B对A的力大小相等、方向相反。若相互作用持续Δt，则B受到的冲量为FAB Δt，A受到的冲量为FBA Δt。因FAB = -FBA，两冲量大小相等、符号相反。冲量等于动量变化，故ΔpB = -ΔpA，即ΔpA + ΔpB = 0。因此，在没有外力的情况下，系统的总动量保持不变。
The Jan22 mark scheme emphasised the need to mention 'isolated system' or 'no external forces' and to explicitly relate impulse to momentum change. This derivation is elegant in its simplicity and links three of Newton's laws.
2022年1月评分标准强调需提及“孤立系统”或“无外力”，并明确将冲量与动量变化挂钩。这一推导简单优美，串联了牛顿三定律。
4. Elastic Potential Energy in a Spring | 弹簧的弹性势能
For a spring obeying Hooke' he extension x: F = kx. The work done in stretching the spring is the area under the force– e mark schemes also accept the derivation through average force: as the force rises linearly from 0 to F, the average force during stretching is ½F, so work done = average force × displacement = ½F x.
对于遵从胡克定律的弹簧，力F与伸长量x成正比：F = kx。拉伸弹簧时所做的功等于力–伸长图下方的面积，为一个三角形。因此弹性势能存储量为½ × 力 × 伸长量 = ½ F x。代入F = kx得到E = ½ kx²。部分评分标准也接受通过平均力进行推导：由于力从0线性增长至F，拉伸过程中的平均力为½F，故做功 = 平均力 × 位移 = ½F x。
The Jan22 scheme required students to note the assumption that the spring's elastic limit is not exceeded, ensuring that Hooke's law is valid and that energy is stored elastically.
2022年1月评分标准要求学生指明假定弹簧未超过弹性极限，确保胡克定律成立且能量以弹性势能形式存储。
5. Young Modulus in Materials | 材料的杨氏模量
tio of stress to strain within the limit of proportionality: E = σ/ε = (F/A) / (ΔL/L) = F L / (A ΔL). This derivation is straightforward but requires careful definition of the quantities. In an exam, you might be asked to derive the expression for the gradient of a stress–strain graph or to calculate E from a force– graph. Then the gradient is force/extension = E A / L, leading to E = (gradient × L) / A.
应力定义为单位横截面积上的力，σ = F/A。应变是长度的分数变化，ε = ΔL/L。杨氏模量E是在比例极限内应力与应变的比值：E = σ/ε = (F/A) / (ΔL/L) = F L / (A ΔL)。这个推导直接，但需要细心定义各量。考试中可能要求你推导弹力–伸长图斜率的表达式，或从力–伸长图计算E，此时斜率 = 力/伸长量 = E A / L，从而E = (斜率 × L) / A。
The Jan22 mark scheme looked for correct definitions of stress and strain and the ability to rearrange the formula to find any unknown. It also stressed that the area should be the original cross-sectional area of the wire.
2022年1月评分标准要求正确给出应力和应变的定义，并能将公式变形求解未知量。同时强调面积应为导线原始横截面积。
6. Double‑Slit Fringe Spacing Formula | 双缝干涉条纹间距公式
In Young's double‑ onstructive interference (bright fringe), this path difference equals nλ. For small angles, sinθ ≈ tanθ ≈ y/D, where y is the distance from the central maximum to the nth bright fringe and D is the slit‑to‑ /d. The fringe spacing Δy between adjacent bright (or dark) fringes is given by Δy = λD/d.
在杨氏双缝实验中，两缝发出的光波到达屏幕上某点的光程差为d sinθ，其中d为缝距。对于干涉相长（亮纹），程差等于nλ。小角度下sinθ ≈ tanθ ≈ y/D，y是第n级亮纹到中央亮纹的距离，D是缝屏间距。因此d (y/D) = nλ，得到y = nλD/d。相邻亮纹（或暗纹）间距Δy = λD/d。
The Jan22 mark scheme rewarded clear statements of the small‑angle approximation and the condition for interference maxima. Candidates were also asked to explain how the fringe separation changes if the slit spacing is increased or if the screen is moved further away.
2022年1月评分标准对清晰陈述小角度近似以及干涉极大条件给予分数。考生还需解释若增大缝距或将屏幕移远，条纹间距如何变化。
7. Resistivity and Resistance of a Wire | 电阻率与导线电阻
istance R is directly proportional to its length L and inversely proportional to its cross‑ esistance of a unit cube of the material. In the Jan22 scheme, students were expected to describe how to determine ρ experimentally by measuring resistance, length, and diameter, and then using R = ρL/A. This formula can also be derived by considering the microscopic model of conduction, but at AS level the empirical relationship is sufficient.
对于均匀金属导体，电阻R与其长度L成正比，与横截面积A成反比。引入比例常数ρ得R = ρL/A。推导始于将电阻率定义为材料单位立方体的电阻。在2022年1月评分标准中，要求学生描述如何通过测量电阻、长度和直径，使用R = ρL/A来实验测定ρ。该公式也可通过导电的微观模型推导，但在AS阶段，经验关系即可。
Examiners looked for correct unit conversions and awareness that resistivity depends on temperature. The mark scheme also highlighted the need to measure the diameter in several places to calculate a mean area.
考官关注单位换算是否正确，以及是否知晓电阻率依赖于温度。评分标准还强调需在多个位置测量直径以求平均截面积。
8. Internal Resistance and Terminal Potential Difference | 内阻与路端电压
y is dissipated inside the source. Conservation of energy demands E = I R + I r, or E = V + I r, where V is the terminal potential difference across the load R. Rearranging gives V = E - I r. The Jan22 mark scheme required candidates to derive this from the definition of E and the energy transfers, often in the context of a circuit experiment where V and I are measured for a variable load.
实际电源具有内阻r。电动势E是每单位电荷提供的能量；当电流I流动时，一部分能量在电源内部耗散。能量守恒要求E = I R + I r，或E = V + I r，其中V是负载R两端的路端电压。整理得V = E - I r。2022年1月评分标准要求考生从E的定义和能量传递出发推导此式，常结合可变负载下测量V、I的实验情境。
The graph of V against I yields a straight line with gradient -r and y‑intercept E. Candidates were expected to interpret such graphs and find internal resistance correctly.
V–I图是一条斜率为-r、截距为E的直线。评分标准期望考生能够解读此类图像并正确求出内阻。
9. Maximum Power Transfer Theorem | 最大功率定理
tiate P with respect to R and set the derivative to zero, or simply use the fact that the denominator is minimised when R = r (by completing the square or the AM‑ creases from zero, power rises to a peak when R = r and then falls. The maximum power is Pmax = E²/(4r).
输送到外负载R的功率为P = I²R。利用I = E/(R + r)，可得P = E²R/(R + r)²。要找出最大功率条件，可对P关于R求导并令导数为零，或利用R = r时分母最小（通过配方或均值不等式）。2022年1月评分标准接受图像论证：随着R从零增大，功率上升，在R = r时达到峰值，随后下降。最大功率Pmax = E²/(4r)。
ces. The mark scheme explicitly rewarded stating that when R = r the efficiency is only 50%.
此推导常出现在有关电源效率的题目中。评分标准明确奖励指出当R = r时，效率仅为50%。
10. Snell's Law from Wave Speed | 用波速推导斯涅耳定律
a plane wave incident at an angle θ₁ to the normal; the wave speed in medium 1 is v₁ and in medium 2 is v₂. In a given time interval, a wavefront travels distances v₁t in medium 1 and v₂t in medium 2 along the ray directions. Geometry of the triangles gives sinθ₁ = (v₁t)/d and sinθ₂ = (v₂t)/d, where d is the distance along the boundary. Hence sinθ₁/sinθ₂ = v₁/v₂. Defining the refractive index n as the ratio of the speed of light in vacuum to that in the medium, n = c/v, we obtain n₁ sinθ₁ = n₂ sinθ₂.
波穿越两介质分界面时，波速改变但频率不变，波前发生偏转。考虑一平面波以入射角θ₁射向法线；介质1中的波速为v₁，介质2中为v₂。在给定时间间隔内，一波前沿射线方向在介质1中走过v₁t，在介质2中走过v₂t。由几何三角形得sinθ₁ = (v₁t)/d，sinθ₂ = (v₂t)/d，其中d是沿分界面的距离。因此sinθ₁/sinθ₂ = v₁/v₂。定义折射率n为真空中光速与介质中光速之比即n = c/v，可得n₁ sinθ₁ = n₂ sinθ₂。
The Jan22 mark scheme placed importance on clear diagrams and labelling of angles. The derivation assumes the boundary is smooth and that the ray model is valid.
2022年1月评分标准重视清晰的图示和角度标注。推导假设界面光滑且射线模型成立。
11. Additional Derivation: Centripetal Acceleration Formula (Optional Depth) | 附加推导：向心加速度公式（可选加深）
ving at constant speed v in a circle of radius r. In a short time Δt, the velocity vector changes direction but not magnitude. The change in velocity points toward the centre. The triangle of velocities is similar to the triangle of displacements, so Δv/v = Δs/r. Dividing by Δt gives acceleration a = Δv/Δt = (v/r)(Δs/Δt) = v²/r. Using v = ωr gives a = ω²r.
虽然向心加速度推导在部分考试大纲中属于单元4，但某些AS课程要求定性推导a = v²/r或a = ω²r。考虑物体沿半径为r的圆周以恒定速率v运动。在短时间Δt内，速度向量方向改变但大小不变，速度变化指向圆心。速度三角形与位移三角形相似，故Δv/v = Δs/r。除以Δt得加速度a = Δv/Δt = (v/r)(Δs/Δt) = v²/r。结合v = ωr得a = ω²r。
The Jan22 scheme accepted an explanation based on the geometry of the vectors, and rewarded candidates who stated the assumption that the speed is constant (uniform circular motion).
2022年1月评分标准接受基于矢量几何的解释，并且对声明速率恒定（匀速圆周运动）的考生给予加分。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图