Forces and Motion | 力与运动考点精讲

📚 Forces and Motion | 力与运动考点精讲

Understanding forces and motion is fundamental to physics and forms a core component of the CCEA GCE Science specification. This article summarises the key concepts, essential equations and common pitfalls to help you build a solid foundation and tackle exam questions with confidence.

理解力与运动是物理学的基础，也是 CCEA GCE 科学教学大纲的核心内容。本文总结关键概念、必备方程和常见误区，帮助你打下坚实基础，自信应对考题。

1. Scalars and Vectors | 标量与矢量

A scalar quantity has magnitude only. Typical examples include mass, speed, distance, energy and temperature.

标量只有大小。常见例子有质量、速率、路程、能量和温度。

A vector quantity has both magnitude and direction. Examples include displacement, velocity, acceleration, force and momentum. Arrows are used to represent vectors, with the length showing magnitude and the arrowhead indicating direction.

矢量既有大小又有方向。例子包括位移、速度、加速度、力和动量。通常用箭头表示矢量，长度表示大小，箭头指向表示方向。

When adding vectors, you must take direction into account. For collinear vectors, simple addition or subtraction works. For perpendicular vectors, use Pythagoras’ theorem and trigonometry to find the resultant.

矢量相加时，必须考虑方向。对于共线矢量，直接相加或相减；对于正交矢量，使用勾股定理和三角关系求合矢量。

Resolving a vector into two perpendicular components is a critical skill. For a force F at angle θ, the horizontal component is F cos θ and the vertical component is F sin θ, measured from the horizontal.

将矢量分解为两个相互垂直的分量是一项关键技能。对于与水平方向夹角为 θ 的力 F，水平分量为 F cos θ，竖直分量为 F sin θ（从水平面量起）。

2. Kinematics Equations for Uniform Acceleration | 匀加速运动学方程

When acceleration is constant, four kinematic equations (often called SUVAT equations) relate displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t).

当加速度恒定时，四个运动学方程（常称 SUVAT 方程）将位移 s、初速度 u、末速度 v、加速度 a 和时间 t 联系起来。

v = u + at

Equation 1: final velocity equals initial velocity plus acceleration multiplied by time.

方程一：末速度等于初速度加上加速度乘以时间。

s = ut + ½ at²

Equation 2: displacement equals initial velocity times time plus half the acceleration times time squared.

方程二：位移等于初速度乘时间加二分之一加速度乘时间的平方。

v² = u² + 2as

Equation 3: the square of final velocity equals square of initial velocity plus twice acceleration times displacement.

方程三：末速度的平方等于初速度的平方加二倍加速度乘位移。

s = ½ (u + v) t

Equation 4: displacement equals average velocity multiplied by time.

方程四：位移等于平均速度乘时间。

Always define a positive direction and use consistent signs for vectors. For vertical motion under gravity, acceleration a becomes g ≈ 9.81 m s⁻² downwards; if upward is positive, use a = −9.81 m s⁻².

务必规定正方向，所有矢量符号保持一致。在重力作用下的竖直运动中，加速度 a 取 g ≈ 9.81 m s⁻² 方向向下；若向上为正，则 a = −9.81 m s⁻²。

3. Newton’s Three Laws of Motion | 牛顿三定律

First Law – Inertia: An object will remain at rest or in uniform motion in a straight line unless acted upon by a net external force.

第一定律（惯性定律）：除非受到净外力作用，物体将保持静止或匀速直线运动状态。

Second Law – Acceleration: The net force on an object is equal to its mass multiplied by its acceleration, F = ma. This law quantifies how a resultant force changes motion.

第二定律（加速度定律）：物体所受合力等于其质量乘以加速度，F = ma。该定律定量描述了合力如何改变运动。

Third Law – Action–Reaction: If body A exerts a force on body B, then body B exerts an equal and opposite force on body A. These forces act on different objects and do not cancel.

第三定律（作用与反作用）：若物体 A 对物体 B 施加力，则物体 B 同时对物体 A 施加大小相等、方向相反的力。这两个力作用在不同物体上，不会抵消。

In calculations, always identify all forces, draw a free-body diagram and apply F_net = ma component by component. Remember that mass m is a scalar, while acceleration and force are vectors.

计算时，务必找出所有力，画出受力图，并按分量应用 F_net = ma。牢记质量 m 是标量，而加速度和力是矢量。

4. Types of Forces and Free-Body Diagrams | 力的类型与受力图

Weight W acts downwards toward the centre of the Earth and equals mg, where g is the gravitational field strength (9.81 N kg⁻¹).

重力 W 方向竖直向下指向地心，大小等于 mg，其中 g 为重力场强度（9.81 N kg⁻¹）。

Normal reaction force N is perpendicular to the contact surface and counteracts the component of weight pressing into the surface.

支持力 N 垂直于接触面，抵消物体压向接触面的重力分量。

Friction f opposes relative motion (or the tendency of motion) between two surfaces in contact. The maximum static friction is given by f_s ≤ μ_s N, and kinetic friction by f_k = μ_k N.

摩擦力 f 阻碍两个接触面的相对运动（或运动趋势）。最大静摩擦力为 f_s ≤ μ_s N，滑动摩擦力为 f_k = μ_k N。

Tension is the force transmitted through a string, rope or cable when it is pulled tight. It pulls equally on the objects at both ends, assuming a massless, inextensible string.

张力是通过绳子、绳索或缆绳拉紧时传递的力。假设绳子轻质且不可伸长，它对两端物体的拉力大小相等。

To draw a free-body diagram, isolate the object, represent it as a point and draw all forces as arrows. Do not include forces exerted by the object on its surroundings.

画受力图时，将物体隔离出来，用一个点表示，并画出所有作用于该物体的力（箭头）。不要包含该物体对周围物体施加的力。

5. Momentum and Impulse | 动量与冲量

Linear momentum p is a vector defined as p = mv, where m is mass and v is velocity. The unit is kg m s⁻¹.

动量 p 是矢量，定义为 p = mv，其中 m 为质量，v 为速度。单位为 kg m s⁻¹。

Impulse J is the product of average force and the time for which it acts: J = F Δt. Impulse equals the change in momentum: F Δt = Δp.

冲量 J 是平均力与作用时间的乘积：J = F Δt。冲量等于动量的变化：F Δt = Δp。

The principle of conservation of momentum states that in a closed system with no external forces, the total momentum before an interaction equals the total momentum after.

动量守恒定律指出，在没有外力的封闭系统中，相互作用前的总动量等于相互作用后的总动量。

Collisions can be elastic (both momentum and kinetic energy conserved) or inelastic (momentum conserved, kinetic energy not conserved). In a perfectly inelastic collision, objects stick together after impact.

碰撞可分为弹性碰撞（动量和动能均守恒）和非弹性碰撞（动量守恒，动能不守恒）。在完全非弹性碰撞中，物体碰撞后粘在一起运动。

6. Work, Power and Energy | 功、功率与能量

Work is done when a force moves an object through a displacement. Work W = F d cos θ, where θ is the angle between the force and displacement. The unit is the joule (J).

当力使物体发生位移时，力做功。功 W = F d cos θ，其中 θ 为力与位移的夹角。单位为焦耳（J）。

Power P is the rate of doing work: P = W / t = F v, where v is velocity when force and velocity are parallel. Unit: watt (W).

功率 P 是做功的快慢：P = W / t = F v，当力与速度方向相同时适用。单位：瓦特（W）。

Kinetic energy E_k = ½ m v² is the energy an object possesses due to its motion. Gravitational potential energy E_p = m g h is the energy stored due to an object’s height above a reference level.

动能 E_k = ½ m v² 是物体由于运动而具有的能量。重力势能 E_p = m g h 是物体因离参考面的高度而储存的能量。

The work–energy principle states that the net work done on an object equals its change in kinetic energy: W_net = ΔE_k.

功能原理指出，合力对物体做的功等于其动能的变化量：W_net = ΔE_k。

7. Conservation of Mechanical Energy | 机械能守恒

When only conservative forces (gravity, elastic spring force) act, the total mechanical energy E = E_k + E_p remains constant.

当只有保守力（重力、弹簧弹力）做功时，总机械能 E = E_k + E_p 保持不变。

This can be written as ½ m v₁² + m g h₁ = ½ m v₂² + m g h₂ for a system where gravity is the only force doing work. This relationship is extremely useful for solving motion along curves and slopes.

对于仅重力做功的系统，可写为 ½ m v₁² + m g h₁ = ½ m v₂² + m g h₂。该关系式在求解沿曲线和斜面运动时非常有用。

In the presence of non-conservative forces such as friction, the change in mechanical energy equals the work done by those forces (typically negative, converting mechanical energy to heat).

当存在摩擦力等非保守力时，机械能的变化等于这些力做的功（通常为负，机械能转化为内能）。

8. Projectile Motion | 抛体运动

A projectile follows a parabolic path under constant gravitational acceleration, ignoring air resistance. The horizontal and vertical motions are independent.

忽略空气阻力时，抛体在恒定重力加速度下沿抛物线轨迹运动。水平与竖直运动彼此独立。

Horizontally, velocity u_x = u cos θ remains constant because there is no acceleration. The horizontal displacement is x = u_x t.

水平方向上，速度 u_x = u cos θ 恒定，因为没有加速度。水平位移为 x = u_x t。

Vertically, motion is governed by the equations of uniform acceleration with a = −g (if upward is positive). The vertical velocity at time t is v_y = u sin θ − g t, and vertical displacement is y = (u sin θ) t − ½ g t².

竖直方向上，运动遵循匀加速方程，加速度 a = −g（向上为正）。t 时刻竖直速度 v_y = u sin θ − g t，竖直位移 y = (u sin θ) t − ½ g t²。

The time of flight, maximum height and range can all be derived from these components. For symmetric level-ground projectiles, range R = (u² sin 2θ)/g and maximum height H = (u² sin² θ)/(2g).

飞行时间、最大高度和射程均可从上述分量方程推导得出。对于对称落地（起落点等高）的抛体，射程 R = (u² sin 2θ)/g，最大高度 H = (u² sin² θ)/(2g)。

9. Circular Motion | 圆周运动

An object moving in a circle at constant speed is undergoing uniform circular motion. Although speed is constant, velocity is changing due to continuous change in direction, so there is a centripetal acceleration directed towards the centre.

物体以恒定速率做圆周运动称为匀速圆周运动。虽然速率不变，但由于方向不断改变，速度在变化，因此存在指向圆心的向心加速度。

The magnitude of centripetal acceleration is a = v² / r = ω² r, where v is the tangential speed, r is the radius and ω is the angular speed (ω = 2π / T = 2π f).

向心加速度的大小为 a = v² / r = ω² r，其中 v 为线速度，r 为半径，ω 为角速度（ω = 2π / T = 2π f）。

The net force required for circular motion is the centripetal force, F = m v² / r = m ω² r, always directed towards the centre. This force is not an extra force but is provided by tension, gravity, friction or normal reaction depending on the situation.

圆周运动所需的合力为向心力，F = m v² / r = m ω² r，始终指向圆心。该力并非额外力，而是根据情境由拉力、重力、摩擦力或支持力提供。

Published by TutorHao | Science Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Forces and Motion | 力与运动 考点精讲