From Moving to Stationary | 从运动到静止

📚 From Moving to Stationary | 从运动到静止

When a train brakes into a station, a ball rolls to rest, or a crumple zone stops a crash, an object is changing from moving to stationary. In CIE A-Level Physics, this process links kinematics, Newton’s laws, work and energy, momentum and impulse, and practical safety engineering. This article explains the physics of deceleration, stopping distance, and impact protection in a clear, exam-focused way.

当一列火车刹车进站、一个球滚到停下，或者碰撞中溃缩区让车停下时，物体正在从运动变为静止。在 CIE A-Level 物理中，这一过程将运动学、牛顿定律、功与能量、动量与冲量以及实际安全工程联系在一起。本文以清晰、贴合考点的方式解释减速、停车距离和碰撞保护的物理原理。

1. Kinematics of Deceleration | 减速运动学

When an object slows down uniformly from an initial speed u to rest, its final velocity is v = 0. The acceleration is negative, so it is often described as a deceleration or retardation. For constant acceleration, the key equation is v² = u² + 2as, where s is the displacement while stopping.

当物体从初速度 u 均匀减速到静止时，其末速度 v = 0。加速度为负，因此通常称为减速度或负加速度。对于匀加速运动，关键方程是 v² = u² + 2as，其中 s 是停止过程中的位移。

Setting v = 0 gives the stopping distance in terms of the initial speed and the deceleration magnitude a:

令 v = 0，可得到用初速度和减速度大小 a 表示的停车距离：

0 = u² − 2as ⇒ s = u² ÷ (2a)

This relation shows that stopping distance is proportional to the square of the initial speed. If a car doubles its speed, the minimum braking distance becomes four times larger for the same deceleration. On a velocity-time graph, the area under the sloping line gives the displacement, and the gradient gives the negative acceleration.

这一关系表明，停车距离与初速度的平方成正比。如果汽车速度变为原来的两倍，在相同减速度下，最小制动距离会变为原来的四倍。在速度-时间图中，斜线下方的面积表示位移，斜率表示负加速度。

2. Newton’s Second Law and Net Braking Force | 牛顿第二定律与净制动力

To make a moving object stationary, a resultant force must act opposite to its direction of motion. Newton’s second law gives F = ma, where F is the net force, m is the mass, and a is the acceleration. During braking, the net force often comes from friction between the tyres and the road, from brake pads pressing on discs, or from a combination of drag and rolling resistance.

要使运动物体静止，必须有一个与其运动方向相反的合力。牛顿第二定律给出 F = ma，其中 F 是净力，m 是质量，a 是加速度。在制动过程中，净力通常来自轮胎与路面之间的摩擦、刹车片压紧制动盘产生的力，或空气阻力与滚动阻力的组合。

If the total braking force is constant, the deceleration is a = F ÷ m. Therefore a heavier vehicle requires a larger braking force to achieve the same deceleration. However, if the braking force scales with the vehicle’s weight, mass can cancel out of the stopping distance equation – a point examined in Section 8.

如果总制动力是恒定的，则减速度 a = F ÷ m。因此，较重的车辆需要更大的制动力才能获得相同的减速度。然而，如果制动力随车辆重量成比例增大，质量会在停车距离方程中抵消——第 8 节将分析这一点。

In free-body diagrams for a braking car, the vertical forces are the weight mg downwards and the normal contact force R upwards. The horizontal resultant is the braking force opposing motion, so the acceleration vector points backwards. Students should clearly label the direction of positive displacement when applying signs.

在制动汽车的受力图中，竖直方向有力：向下的重力 mg 和向上的法向接触力 R。水平方向的合力是与运动方向相反的制动力，因此加速度矢量指向后方。学生在应用正负号时应清楚标明正位移方向。

3. Friction and Types of Braking | 摩擦与制动类型

Friction between the tyres and the road provides the external force that slows the vehicle. The maximum friction force is proportional to the normal contact force R and the coefficient of friction μ between the two surfaces: F_friction ≤ μR. On a level road, R = mg, so the maximum friction is μmg.

轮胎与路面之间的摩擦提供使车辆减速的外力。最大摩擦力与法向接触力 R 和两表面间的摩擦系数 μ 成正比：F_friction ≤ μR。在水平路面上，R = mg，因此最大摩擦力为 μmg。

If the wheels lock during heavy braking, the tyres slide over the road. Kinetic friction then acts, and the driver loses steering control. Anti-lock braking systems (ABS) prevent locking by rapidly adjusting brake pressure, keeping the tyres near the point of maximum static friction while still rolling. This improves both braking force and directional control.

如果急刹车时车轮抱死，轮胎就会在路面上滑动。此时作用的是动摩擦，驾驶员会失去转向控制。防抱死制动系统（ABS）通过快速调节制动压力来防止抱死，使轮胎保持在接近最大静摩擦且仍在滚动的状态。这既提高了制动力，又保持了方向控制。

Inside the vehicle, disc brakes and drum brakes use friction to convert kinetic energy into heat. Brake pads are pressed against a rotating disc, producing a frictional torque that opposes wheel rotation. Overheating can reduce friction – an effect called brake fade – so high-performance brakes use ventilated discs to dissipate heat.

在车辆内部，盘式制动器和鼓式制动器利用摩擦将动能转化为热能。刹车片被压紧在旋转的制动盘上，产生阻碍车轮转动的摩擦力矩。过热会降低摩擦力——这一现象称为制动热衰退——因此高性能制动器使用通风盘来散热。

4. Work Done and Kinetic Energy Transfer | 做功与动能转化

From the energy perspective, stopping a moving object means removing its kinetic energy. The work-energy principle states that the work done by the resultant force equals the change in kinetic energy. If an object of mass m slows from speed u to rest, the change in kinetic energy is 0 − ½mu² = −½mu².

从能量角度看，使运动物体停止意味着消除其动能。功-能原理指出，合力所做的功等于动能的变化。如果质量为 m 的物体从速度 u 减速到静止，动能变化为 0 − ½mu² = −½mu²。

The braking force F does negative work over the stopping distance s, so the work done is −Fs. Equating the work done to the kinetic energy change gives:

制动力 F 在停车距离 s 上做负功，因此做功为 −Fs。令做功等于动能变化，可得：

Fs = ½mu² ⇒ s = mu² ÷ (2F)

This is the same relationship obtained from kinematics when a = F ÷ m, confirming that Newton’s laws and energy methods are consistent. In traditional friction brakes, most of this kinetic energy is transferred to the internal energy of the brake discs and pads, raising their temperature.

这与用运动学方法在 a = F ÷ m 时得到的关系相同，证明牛顿定律与能量方法是自洽的。在传统摩擦制动器中，大部分动能转化为制动盘和刹车片的内能，使其温度升高。

5. Momentum and Impulse in Stopping | 动量与冲量

Momentum is the product of mass and velocity, p = mv. When an object stops, its momentum changes from mu to zero. The change in momentum is Δp = 0 − mu = −mu, with the negative sign showing that the change opposes the original motion.

动量是质量与速度的乘积，p = mv。当物体停止时，其动量从 mu 变为零。动量变化为 Δp = 0 − mu = −mu，负号表示变化方向与原始运动方向相反。

Impulse is the product of the resultant force and the time for which it acts: impulse = FΔt. The impulse-momentum theorem states that impulse equals the change in momentum:

冲量是合力与其作用时间的乘积：impulse = FΔt。冲量-动量定理指出，冲量等于动量的变化：

FΔt = mu

Therefore the average stopping force is F = mu ÷ Δt. A shorter stopping time produces a larger average force for the same momentum change. This is why a sudden impact is much more damaging than a controlled stop over a longer time.

因此平均制动力为 F = mu ÷ Δt。对于相同的动量变化，停止时间越短，产生的平均力越大。这就是为什么突然撞击比长时间受控停车造成的破坏大得多。

6. Force-Time Graphs and Crumple Zones | 力-时间图与溃缩区

A force-time graph shows how the force acting during a collision or stop varies with time. The area under the graph is equal to the impulse, which is the change in momentum. In a real crash, the force rises rapidly to a sharp peak and then falls, giving a large maximum force over a very short time.

力-时间图显示碰撞或停止过程中作用力随时间的变化。图线下方的面积等于冲量，也就是动量的变化。在真实碰撞中，力会迅速上升到尖锐的峰值然后下降，在极短时间内产生很大的最大力。

Crumple zones at the front and rear of a car are designed to deform plastically in a controlled way. By increasing the time over which the car comes to rest, the same impulse is achieved with a lower average force. The peak force is also reduced because the graph becomes wider and flatter while the area stays constant.

汽车前后部的溃缩区被设计为以受控方式发生塑性变形。通过延长汽车停下所需的时间，可以在相同冲量下获得较小的平均力。峰值力也会降低，因为图线变得更宽更平，而面积保持不变。

A rigid chassis does not absorb much energy and stops almost instantly, producing a huge force spike. In contrast, a modern car with a crumple zone extends the collision time from about 0.1 s to 0.3 s or more, reducing the average force by a factor of three for the same momentum change.

刚性车身几乎不吸收能量，几乎瞬间停下，产生巨大的力尖峰。相比之下，带溃缩区的现代汽车可将碰撞时间从约 0.1 s 延长到 0.3 s 或更长，在相同动量变化下，平均力可降低到约三分之一。

7. Thinking, Braking and Stopping Distances | 反应距离、制动距离与停车距离

The total stopping distance of a vehicle has two parts. The thinking distance is the distance travelled while the driver reacts before applying the brakes. The braking distance is the distance travelled after the brakes are applied until the vehicle stops. The total stopping distance is their sum.

车辆的总停车距离由两部分组成。反应距离是驾驶员在踩刹车前反应时间内行驶的距离。制动距离是刹车开始作用后到车辆停止所行驶的距离。总停车距离是两者之和。

The thinking distance is given by speed multiplied by reaction time: s_thinking = u t_reaction. A typical reaction time is 0.6-1.0 s. The braking distance is given by s_braking = u² ÷ (2a), assuming constant deceleration. The total stopping distance is therefore:

反应距离等于速度乘以反应时间：s_thinking = u t_reaction。典型反应时间为 0.6-1.0 s。制动距离由 s_braking = u² ÷ (2a) 给出，假设减速度恒定。因此总停车距离为：

s_total = u t_reaction + u² ÷ (2a)

The table below shows typical values for a car with a reaction time of 0.7 s and a braking deceleration of 6.0 m/s².

下表显示反应时间为 0.7 s、制动减速度为 6.0 m/s² 的汽车的典型数值。

Speed u (m/s)	Thinking distance (m)	Braking distance (m)	Total stopping distance (m)
10	7.0	8.3	15.3
20	14.0	33.3	47.3
30	21.0	75.0	96.0

Notice that doubling the speed from 20 m/s to 40 m/s would quadruple the braking distance, while the thinking distance only doubles. High speed therefore has a disproportionate effect on stopping distance.

注意，速度从 20 m/s 加倍到 40 m/s 会使制动距离变为四倍，而反应距离只加倍。因此，高速对停车距离的影响不成比例地大。

8. Factors Affecting Braking Distance | 影响制动距离的因素

Braking distance depends on the initial speed, the braking force, the road surface, the tyre condition, the brake condition, and the gradient. Wet or icy roads reduce the coefficient of friction μ, so the maximum braking force decreases and the braking distance increases. Worn tyres or brakes have the same effect.

制动距离取决于初速度、制动力、路面状况、轮胎状况、制动器状况和坡度。湿滑或结冰的路面会降低摩擦系数 μ，因此最大制动力减小，制动距离增加。磨损的轮胎或制动器也有同样的影响。

Speed: braking distance ∝ u² if deceleration is constant – 速度：若减速度恒定，制动距离 ∝ u²
Friction coefficient: lower μ on ice or wet roads gives a smaller maximum deceleration – 摩擦系数：冰面或湿滑路面 μ 较小，最大减速度也较小
Mass: if braking force is proportional to weight, mass cancels; if force is fixed, greater mass increases distance – 质量：若制动力与重量成正比，质量会抵消；若力固定，质量越大距离越大
Slope: braking uphill is shorter, braking downhill is longer because a component of weight helps or opposes stopping – 坡度：上坡制动距离较短，下坡制动距离较长，因为重力的分量会帮助或阻碍停车

The mass point is a common exam trap. With friction braking, the maximum friction is μmg, so the maximum deceleration is μg and the braking distance is s = u² ÷ (2μg), independent of mass. However, if a question assumes a constant braking force F, then a larger mass gives a smaller deceleration and a longer braking distance.

质量这一点是常见的考试陷阱。在摩擦制动中，最大摩擦力为 μmg，因此最大减速度为 μg，制动距离为 s = u² ÷ (2μg)，与质量无关。但如果题目假设制动力 F 恒定，那么质量越大，减速度越小，制动距离越长。

9. Regenerative and Engine Braking | 再生制动与发动机制动

In conventional braking, kinetic energy is wasted as heat in the brake discs and pads. Regenerative braking, used in hybrid and electric vehicles, uses the electric motor as a generator. The motor converts kinetic energy into electrical energy, which is stored in the battery for later use.

在传统制动中，动能被浪费为制动盘和刹车片中的热量。再生制动用于混合动力和电动汽车，它把电动机用作发电机。电动机将动能转化为电能，并储存在电池中供以后使用。

Regenerative braking does not bring a vehicle completely to rest in most systems; friction brakes are still needed at low speed. However, it reduces brake wear and improves overall energy efficiency. From the physics viewpoint, the motor provides a backwards force that does negative work, removing kinetic energy without relying only on friction heating.

大多数系统中，再生制动不能使车辆完全停下；低速时仍需要摩擦制动。但它减少了制动器磨损，提高了整体能源效率。从物理角度看，电动机提供一个向后的力做负功，在不只依靠摩擦生热的情况下减少动能。

Engine braking occurs when a driver downshifts and releases the accelerator. The engine’s compression strokes absorb energy, slowing the car without using the brake pads. This is useful on long downhill slopes because it prevents overheating of the friction brakes.

发动机制动发生在驾驶员降挡并松开油门时。发动机的压缩冲程吸收能量，在不使用刹车片的情况下使汽车减速。这在长下坡路段很有用，因为它可以防止摩擦制动器过热。

10. Safety Systems: Seatbelts and Airbags | 安全系统：安全带与安全气囊

Seatbelts and airbags are designed to protect occupants by increasing the time over which a person’s momentum changes. A seatbelt locks during sudden deceleration and stretches slightly, so the torso slows with the car rather than continuing forward at the original speed. The stretching increases the stopping time and reduces the average force on the chest.

安全带和安全气囊旨在通过延长人体动量变化的时间来保护乘员。安全带在突然减速时锁止并略微拉伸，使躯干随车减速，而不是以原速继续前冲。拉伸增加了停止时间，减小了作用在胸部上的平均力。

An airbag inflates within about 0.03 s and provides a soft cushion. It spreads the force over a larger area of the body, reducing pressure and the risk of serious injury. The impulse is fixed because the change in momentum of the occupant is fixed, so increasing the collision time reduces the average force.

安全气囊在约 0.03 s 内充气展开，提供柔软的缓冲。它将力分散到身体更大的面积上，降低压强和重伤风险。由于乘员的动量变化是固定的，冲量也是固定的，所以延长碰撞时间可以降低平均力。

Head restraints prevent whiplash by stopping the head from moving backwards relative to the body in a rear impact. Crumple zones, seatbelts, airbags, and padded interiors all apply the same impulse-momentum principle to reduce injury.

头枕通过防止后碰时头部相对于身体向后甩动来避免颈部扭伤。溃缩区、安全带、安全气囊和软质内饰都应用了相同的冲量-动量原理来减少伤害。

11. Experimental Methods and Data Analysis | 实验方法与数据分析

A simple school experiment can investigate how stopping distance depends on speed. A toy car or trolley is released from different heights on a ramp and allowed to roll onto a horizontal surface with a rough material. The distance travelled before stopping is measured. Alternatively, light gates can record the initial speed u, and a motion sensor can record the stopping displacement s.

一个简单的学校实验可以研究停车距离如何随速度变化。将玩具车或小车从斜坡上不同高度释放，让其滚到铺有粗糙材料的水平面上。测量停止前行驶的距离。也可以使用光门记录初速度 u，使用运动传感器记录停止位移 s。

Plotting s against u² should give a straight line through the origin if the deceleration is constant, because s = u² ÷ (2a). The gradient of the line is 1 ÷ (2a), so the deceleration can be calculated. Students should repeat measurements and calculate uncertainties to evaluate the quality of the data.

如果减速度恒定，绘制 s 与 u² 的关系图应得到一条过原点的直线，因为 s = u² ÷ (2a)。直线的斜率为 1 ÷ (2a)，因此可以计算减速度。学生应重复测量并计算不确定度，以评估数据质量。

Another common practical uses a force sensor and motion sensor to record force and velocity during a collision. The area under the force-time graph can be compared with the measured change in momentum to verify the impulse-momentum relationship.

另一个常见实验使用力传感器和运动传感器记录碰撞过程中的力和速度。可将力-时间图下的面积与测得的动量变化进行比较，以验证冲量-动量关系。

12. Common Misconceptions and Exam Tips | 常见误区与应试技巧

One misconception is that a heavier vehicle always needs a longer braking distance. This is only true if the braking force is fixed. In normal road conditions, maximum friction is proportional to weight, so mass cancels and the distance depends mainly on speed and friction coefficient. Always check the model stated in the question.

一个常见误区是较重的车辆总是需要更长的制动距离。这只有在制动力固定时才成立。在正常路面条件下，最大摩擦力与重量成正比，因此质量会抵消，制动距离主要取决于速度和摩擦系数。答题时一定要看清题目给出的模型。

Another misconception is that a collision’s force can be reduced by reducing the momentum change. In fact, the momentum change is fixed by the initial speed and final stationary state. Safety devices reduce the average force by increasing the time, not by changing the total impulse.

另一个误区是可以通过减少动量变化来降低碰撞力。实际上，动量变化由初速度和最终静止状态决定，是固定的。安全装置通过延长时间来降低平均力，而不是改变总冲量。

When answering CIE exam questions, define the positive direction, use the correct signs for vectors, and distinguish between thinking distance and braking distance. Quoting s = u² ÷ (2a) and FΔt = mu with clear notation earns credit. If a graph is given, remember that the area under a force-time graph is impulse and the gradient of a velocity-time graph is acceleration.

回答 CIE 考试题时，应定义正方向，正确使用矢量符号，并区分反应距离和制动距离。清楚地写出 s = u² ÷ (2a) 和 FΔt = mu 并标明符号会得到分数。如果题目给出图像，请记住力-时间图下方的面积是冲量，速度-时间图的斜率是加速度。

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From Moving to Stationary | 从运动到静止