A-Level WJEC Science: Forces and Motion Exam Essentials | A-Level WJEC 科学：力与运动考点精讲

📚 A-Level WJEC Science: Forces and Motion Exam Essentials | A-Level WJEC 科学：力与运动考点精讲

Forces and motion form the backbone of classical mechanics in the WJEC A-Level Science specification. This topic links mathematical modelling with real-world phenomena, from the flight of a ball to the orbits of satellites. Mastering these concepts requires a clear understanding of vectors, kinematic equations, Newton’s laws, momentum, energy, and circular motion. This article distils the essential knowledge you need for the exam, with worked examples and common pitfalls to avoid.

力与运动是WJEC A-Level科学大纲中经典力学的支柱。该专题将数学建模与实际现象联系起来，从球的飞行到卫星轨道。要掌握这些概念，需要清晰理解矢量、运动学方程、牛顿定律、动量、能量和圆周运动。本文提炼了考试所需的核心知识，并给出了示例和常见错误避免。

1. Scalars and Vectors | 标量与矢量

Scalars are quantities that have magnitude only. Examples include distance (measured in metres), speed (m/s), mass (kg), and time (s). Vectors possess both magnitude and direction. Displacement, velocity, acceleration, and force are vectors. When adding vectors, the resultant can be found using the parallelogram or triangle method. For a force F at angle θ, the horizontal component is F cos θ and the vertical component is F sin θ.

标量只有大小的量。例如距离（单位米）、速率（m/s）、质量（kg）和时间（秒）。矢量具有大小和方向。位移、速度、加速度和力都是矢量。矢量相加时，可用平行四边形法则或三角形法则求合矢量。对于与水平方向夹角θ的力F，水平分量为F cos θ，竖直分量为F sin θ。

It is essential to resolve vectors into perpendicular components before performing calculations. This skill underpins all of mechanics, especially in free-body diagrams and projectile motion. Always specify direction clearly, using positive and negative signs consistently.

进行计算前，必须将矢量分解为垂直分量。这项技能是整个力学的基础，尤其在受力分析和抛体运动中。始终明确方向，一致地用正负号表示。

2. Equations of Motion | 运动学方程

For uniform acceleration, four key equations relate displacement s, initial velocity u, final velocity v, acceleration a, and time t:

对于匀加速运动，四个核心方程关联位移s、初速度u、末速度v、加速度a和时间t：

v = u + at

s = ut + ½ at²

v² = u² + 2as

s = ½ (u + v) t

These equations apply only when acceleration is constant. For free-fall near Earth’s surface, the acceleration is g = 9.81 m/s² downwards. Always choose a sign convention; for example, take upward as positive so that a = −9.81 m/s² for a rising object.

这些方程仅在加速度恒定时适用。在地表附近的自由落体中，加速度为g = 9.81 m/s²，方向向下。务必选取符号约定；例如取向上为正，则上升物体的a = −9.81 m/s²。

3. Displacement-Time and Velocity-Time Graphs | 位移-时间图与速度-时间图

On a displacement–time graph, the gradient equals velocity. A straight line indicates constant velocity; a curve indicates acceleration. On a velocity–time graph, the gradient gives acceleration, and the area under the graph gives displacement. A horizontal line represents constant velocity, and a sloping line represents uniform acceleration.

在位移-时间图中，斜率等于速度。直线表示匀速；曲线表示加速度。在速度-时间图中，斜率表示加速度，图线下的面积表示位移。水平线代表匀速，倾斜线代表匀加速。

For non-uniform motion, you may need to estimate the area by counting squares or using the trapezium rule. The syllabus expects you to interpret and sketch these graphs, identifying key features such as maximum velocity and total distance travelled.

对于非匀变速运动，你可能需要数格或使用梯形法则估算面积。大纲要求你会解读并绘制这些图像，识别最大速度、总路程等关键特征。

4. Newton’s Laws of Motion | 牛顿运动定律

Newton’s first law: an object remains at rest or in uniform motion unless acted upon by a resultant external force. This defines inertia. Newton’s second law: the resultant force is proportional to the rate of change of momentum. For constant mass, this simplifies to F = ma. Newton’s third law: if body A exerts a force on body B, then body B exerts an equal and opposite force on body A.

牛顿第一定律：除非受到合外力作用，物体将保持静止或匀速直线运动。这定义了惯性。第二定律：合外力与动量变化率成正比。对于质量不变的情况，可简化为F = ma。第三定律：若物体A对物体B施加力，则物体B对A施加等大反向的力。

F = m a

Remember that F is the net force. Always resolve forces and apply Newton’s second law in each perpendicular direction independently. Force pairs from the third law act on different bodies and never cancel out on a single body.

记住F是合力。务必分解力，并在每个正交方向上独立应用牛顿第二定律。第三定律中的力对作用在不同物体上，永远不会在单个物体上抵消。

5. Forces and Free-Body Diagrams | 力与受力分析图

A free-body diagram shows all forces acting on a single object. Common forces include weight (W = mg), normal reaction, friction, tension, and air resistance. Draw arrows representing forces, labelled with their magnitudes or symbols. For equilibrium, the vector sum of forces is zero; the object is either at rest or moving at constant velocity.

受力分析图展示作用在单个物体上的所有力。常见力包括重力（W = mg）、支持力、摩擦力、张力和空气阻力。画箭头表示力，并标注大小或符号。平衡时，力的矢量和为零；物体可能静止或匀速运动。

When dealing with inclined planes, resolve the weight into components parallel and perpendicular to the slope. The parallel component is mg sin θ, and the perpendicular component is mg cos θ. This allows you to apply Newton’s second law along the slope.

处理斜面时，将重力分解为沿斜面和垂直于斜面的分量。平行分量为mg sin θ，垂直分量为mg cos θ。这样就可以沿斜面应用牛顿第二定律。

6. Momentum and Impulse | 动量与冲量

Momentum p is defined as the product of mass and velocity: p = mv. It is a vector measured in kg m/s. Impulse is the effect of a force acting over time, given by J = FΔt. According to Newton’s second law, impulse equals the change in momentum: FΔt = Δp.

动量p定义为质量与速度的乘积：p = mv。它是矢量，单位为kg m/s。冲量是力在一段时间内的作用效果，由J = FΔt给出。根据牛顿第二定律，冲量等于动量的变化：FΔt = Δp。

F Δt = m v − m u

The area under a force–time graph represents impulse. In many exam questions, you will calculate the average force during a collision or the change in velocity from a given impulse.

力-时间图下的面积表示冲量。在许多考题中，你需要计算碰撞过程中的平均力，或根据给定冲量求速度变化。

7. Conservation of Momentum | 动量守恒

In a closed system with no external resultant force, total momentum is conserved. This principle is vital for analysing collisions and explosions. For a collision between two bodies: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. Direction must be taken into account by choosing a positive direction.

在没有合外力的封闭系统中，总动量守恒。这一原理对于分析碰撞和爆炸至关重要。对于两个物体的碰撞：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。需选定正方向以考虑方向。

Collisions can be elastic (kinetic energy conserved) or inelastic (some kinetic energy transformed). In a perfectly inelastic collision, objects stick together and move with a common final velocity. Explosions are the reverse: internal forces push fragments apart, but total momentum remains zero if initially at rest.

碰撞可分为弹性（动能守恒）和非弹性（部分动能转化）。完全非弹性碰撞中，物体粘在一起以共同速度运动。爆炸则相反：内力将碎片推开，但若初始静止则总动量保持为零。

8. Work, Energy and Power | 功、能量与功率

Work done by a constant force is W = Fs cos θ, where θ is the angle between the force and displacement. The work–energy theorem states that the net work done on an object equals its change in kinetic energy: W = ½ m v² − ½ m u². Gravitational potential energy is E_p = mgh, and kinetic energy is E_k = ½ m v².

恒力做功为W = Fs cos θ，其中θ是力与位移的夹角。动能定理指出，合力对物体做的功等于其动能变化：W = ½ m v² − ½ m u²。重力势能为E_p = mgh，动能为E_k = ½ m v²。

Power is the rate of doing work: P = W / t. It can also be expressed as P = Fv for an object moving at constant velocity against a force. Efficiency is the ratio of useful output power to input power, often given as a percentage.

功率是做功的速率：P = W / t。对于以恒定速度对抗力的物体，功率也可表示为P = Fv。效率是有用输出功率与输入功率的比值，常用百分比表示。

9. Circular Motion | 圆周运动

An object moving in a circle at constant speed is accelerating because its direction changes continuously. The centripetal acceleration is directed towards the centre: a = v² / r = ω² r, where ω is the angular speed (ω = Δθ / Δt = 2π / T). The resultant centripetal force is F = m v² / r = m ω² r.

物体以恒定速率做圆周运动时，因方向不断改变而具有加速度。向心加速度指向圆心：a = v² / r = ω² r，其中ω是角速度（ω = Δθ / Δt = 2π / T）。向心力为F = m v² / r = m ω² r。

Examples include a car rounding a bend (friction provides centripetal force), a satellite in orbit (gravity), and a ball on a string. Remember that the centripetal force is not a new type of force but the name given to the resultant force pointing towards the centre.

例子包括汽车转弯（摩擦力提供向心力）、轨道卫星（引力）和绳端小球。记住，向心力不是一种新的力，而是指向圆心的合力名称。

10. Gravitational Forces and Satellite Motion | 万有引力与卫星运动

Newton’s law of universal gravitation states that every mass attracts every other mass with a force: F = G m₁ m₂ / r², where G = 6.67 × 10⁻¹¹ N m² / kg². Near Earth’s surface, this gives the familiar weight W = mg, where g = GM / R² (M and R are Earth’s mass and radius).

万有引力定律指出，任何两个质量都会相互吸引：F = G m₁ m₂ / r²，其中G = 6.67 × 10⁻¹¹ N m² / kg²。在地表附近，由此可得熟悉的重量W = mg，且g = GM / R²（M与R为地球质量和半径）。

For a satellite in a circular orbit, gravitational force supplies the centripetal force: GMm / r² = mv² / r. This leads to orbital speed v = √(GM / r) and period T² = (4π² / GM) r³, which is Kepler’s third law. Geostationary satellites have a period of 24 hours and orbit above the equator.

对于圆形轨道上的卫星，引力提供向心力：GMm / r² = mv² / r。由此可得轨道速率v = √(GM / r) 和周期 T² = (4π² / GM) r³，即开普勒第三定律。地球同步卫星周期为24小时，轨道位于赤道上空。

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A-Level WJEC Science: Forces and Motion Exam Essentials | A-Level WJEC 科学：力与运动 考点精讲