Dynamics Formulas: Summary and Applications | 动力学公式归纳与应用

📚 Dynamics Formulas: Summary and Applications | 动力学公式归纳与应用

Dynamics is the branch of mechanics that explains how forces affect the motion of objects. In A-level, IB and equivalent courses, dynamics questions connect kinematics, momentum, energy and circular or oscillatory motion. Memorising formulas is not enough; you must know when each formula is valid.

动力学是力学中研究力如何影响物体运动的分支。A-level、IB 等国际课程中的动力学题目，往往将运动学、动量、能量以及圆周运动或振动联系在一起。单纯记忆公式并不足够，关键还要掌握每个公式的适用条件。

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1. Newton’s Laws of Motion and Force | 牛顿运动定律与力

Newton’s laws form the foundation of dynamics. The first law defines inertia: if the resultant force on an object is zero, its velocity remains constant. The second law quantifies how a resultant force changes motion:

牛顿运动定律是动力学的基石。第一定律定义惯性：若物体所受合外力为零，则其速度保持不变。第二定律定量描述了合外力如何改变物体运动状态：

ΣF = ma

Here ΣF is the resultant force in newtons (N), m is the mass in kilograms (kg), and a is the acceleration in metres per second squared (m/s²). Near the Earth’s surface, gravity gives a weight W = mg, where g ≈ 9.8 m/s².

式中 ΣF 为合外力（单位 N），m 为质量（单位 kg），a 为加速度（单位 m/s²）。在地球表面附近，物体因重力而受到的重力大小为 W = mg，其中 g ≈ 9.8 m/s²。

Newton’s third law states that if object A exerts a force on object B, object B exerts an equal and opposite force on A. These action-reaction pairs act on different bodies, so they cannot cancel each other on the same object.

牛顿第三定律指出：若物体 A 对物体 B 施加力，则物体 B 必然同时对物体 A 施加大小相等、方向相反的力。这两个作用力与反作用力作用在不同物体上，因此不能在同一物体上相互抵消。

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2. Kinematic Equations for Constant Acceleration | 匀变速直线运动公式

When acceleration is constant, four kinematic equations connect displacement s, initial velocity u, final velocity v, acceleration a and time t.

当加速度恒定时，四个运动学公式将位移 s、初速度 u、末速度 v、加速度 a 和时间 t 联系起来。

v = u + at

s = ½(u + v)t

s = ut + ½at²

v² = u² + 2as

These equations are vector equations. You must choose a positive direction and keep signs consistent. In vertical motion under gravity, the acceleration is a = g = 9.8 m/s² downward, so the sign of g depends on your chosen direction.

以上公式均为矢量式。解题时必须先规定正方向，并始终保持正负号一致。在竖直落体运动中，加速度 a = g = 9.8 m/s²，方向向下；因此 g 的符号取决于你所选的正方向。

Remember that these kinematic equations are only valid for constant acceleration. If acceleration changes, you cannot apply them directly.

请记住，这些运动学公式只适用于匀加速运动。若加速度发生变化，则不能直接套用。

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3. Momentum and Impulse | 动量与冲量

Momentum is a vector quantity defined as the product of mass and velocity. Impulse measures the effect of a force acting over a time interval and equals the change in momentum.

动量是矢量，定义为质量与速度的乘积。冲量描述力在一段时间内产生的累积效果，其大小等于物体动量的变化量。

p = mv, I = FΔt = Δp = mv − mu

For an isolated system, the total momentum is conserved during collisions and explosions. For two bodies colliding head-on along a straight line:

对孤立系统而言，碰撞或爆炸过程中系统总动量守恒。对于两个物体在同一直线上发生对心碰撞，有：

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Momentum conservation applies to all collisions, including inelastic ones. However, kinetic energy is conserved only in perfectly elastic collisions. In a completely inelastic collision, the objects stick together and move with a common velocity.

动量守恒适用于所有碰撞，包括非弹性碰撞。但只有完全弹性碰撞才守恒动能。在完全非弹性碰撞中，两物体粘在一起，以共同速度运动。

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4. Work, Energy and Power | 功、能量与功率

Work is done when a force causes a displacement. For a constant force F acting at an angle θ to the displacement s:

当力使物体发生位移时，力就做了功。对于恒力 F，若它与位移 s 的夹角为 θ，则做功为：

W = Fs cosθ

The most frequently used energy expressions are the kinetic energy of a moving object and the gravitational potential energy near the Earth’s surface:

最常用的能量表达式是运动物体的动能和地球表面附近的重力势能：

KE = ½mv², PE = mgh

The work-energy theorem states that the total work done on an object equals its change in kinetic energy:

动能定理指出：对物体所做的总功等于物体动能的变化量：

W_net = ΔKE = ½mv² − ½mu²

Power is the rate of doing work. For a constant force acting on an object moving at constant velocity v in the direction of the force:

功率是做功的快慢。当恒力 F 作用于物体，且物体沿力的方向以速度 v 匀速运动时：

P = W/t = Fv

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5. Friction and Drag Forces | 摩擦力与阻力

Friction is a contact force that opposes relative motion between surfaces. Static friction prevents an object from starting to move, while kinetic friction acts during sliding.

摩擦力是阻碍物体间相对运动的接触力。静摩擦力阻止物体开始运动，动摩擦力则在物体滑动时起作用。

f_s ≤ μ_s N, f_k = μ_k N

Here N is the normal reaction force. The normal force is perpendicular to the contact surface and is not always equal to the weight of the object. For example, on an inclined plane, N = mg cosθ.

式中 N 为支持力。支持力垂直于接触面，但并不总是等于物体重力。例如在斜面上，N = mg cosθ。

Air resistance and fluid drag oppose motion through a fluid. At low speeds, drag is roughly proportional to velocity; at high speeds, it is proportional to v². A falling object reaches terminal velocity when the upward drag force becomes equal to its weight, giving zero resultant force

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