Momentum and Impulse | 动量与冲量

📚 Momentum and Impulse | 动量与冲量

In Edexcel A-Level Physics, momentum and impulse connect Newton’s laws to real interactions such as collisions, explosions, and safety design. A clear understanding of their vector nature, conservation principles, and force-time relationships is essential for both calculation and explanation questions.

在 Edexcel A-Level 物理中，动量与冲量将牛顿运动定律与碰撞、爆炸和安全设计等真实相互作用联系起来。清楚地理解它们的矢量性质、守恒原理以及力与时间的关系，对计算题和解释题都至关重要。

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1. Defining Linear Momentum | 线动量的定义

Linear momentum p is defined as the product of an object’s mass m and its velocity v:

线动量 p 定义为物体质量 m 与其速度 v 的乘积：

p = m v

Momentum is a vector quantity, so its direction is the same as the direction of velocity. In the International System of Units, momentum is measured in kg m s⁻¹, which is equivalent to N s.

动量是矢量，因此其方向与速度方向相同。在国际单位制中，动量的单位是 kg m s⁻¹，与 N s 等价。

For example, a 1500 kg car travelling at 20 m s⁻¹ due east has momentum p = 1500 × 20 = 30 000 kg m s⁻¹ due east. The magnitude alone is not enough when solving problems involving opposite directions.

例如，一辆 1500 kg 的汽车以 20 m s⁻¹ 向东行驶，其动量 p = 1500 × 20 = 30 000 kg m s⁻¹，方向向东。在涉及相反方向的问题中，仅给出大小是不够的。

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2. Impulse and Change in Momentum | 冲量与动量变化

Impulse J is the product of a resultant force F and the time interval Δt for which it acts:

冲量 J 是合力 F 与其作用时间间隔 Δt 的乘积：

J = F Δt

According to the impulse-momentum relationship, impulse equals the change in momentum of the object:

根据冲量-动量关系，冲量等于物体动量的变化：

J = Δp = m v − m u

Here u is the initial velocity and v is the final velocity. Since velocity is a vector, impulse is also a vector and must be assigned a direction. The unit of impulse is N s, which is identical to kg m s⁻¹.

其中 u 是初速度，v 是末速度。由于速度是矢量，冲量也是矢量，必须规定方向。冲量的单位是 N s，与 kg m s⁻¹ 相同。

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3. Newton’s Second Law in Momentum Form | 牛顿第二定律的动量形式

Newton’s second law was originally expressed in terms of momentum: the resultant force on an object equals the rate of change of its momentum:

牛顿第二定律最初是用动量表述的：作用在物体上的合力等于其动量的变化率：

F = Δp / Δt

For a constant mass, this reduces to F = m a because Δp = m Δv and Δv / Δt = a. However, the momentum form is more general because it can also describe systems where mass changes, such as a rocket ejecting exhaust gases.

当质量不变时，该式简化为 F = m a，因为 Δp = m Δv，且 Δv / Δt = a。但动量形式更普遍，因为它也可以描述质量变化的系统，例如喷出燃气的火箭。

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4. Force-Time Graphs and Impulse | 力-时间图像与冲量

When a force varies during a collision or impact, the impulse delivered is equal to the area under the force-time graph. For a constant force, this area is simply F Δt.

当力在碰撞或撞击过程中变化时，施加的冲量等于力-时间图像下的面积。对于恒力，该面积就是 F Δt。

Many real impacts show a sharp rise and fall in force. The peak force and contact time determine the total area. Exam questions often ask you to estimate impulse by counting squares or using approximate triangular areas.

许多真实撞击的力会迅速上升然后下降。峰值力和接触时间决定了总面积。考试题常要求通过数格或近似三角形面积来估算冲量。

The area under the graph is equal to Δp, not the force at a single instant. A taller peak does not necessarily mean a larger impulse if the contact time is very short.

图像下的面积等于 Δp，而不是某一瞬间的力。如果接触时间很短，较高的峰值并不一定意味着更大的冲量。

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5. Conservation of Linear Momentum | 线动量守恒

The principle of conservation of linear momentum states that in a closed system with no external resultant force, the total momentum before an interaction equals the total momentum after the interaction:

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

m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂

This law applies to collisions and explosions. It is a vector equation, so a positive direction must be chosen and opposite velocities must be given negative signs.

该定律适用于碰撞和爆炸。它是矢量方程，因此必须选取正方向，相反方向的速度必须取负号。

External forces such as friction or gravity can be ignored during the very short time of an explosion or collision, because the internal forces are much larger. This approximation makes conservation valid for many practical problems.

在爆炸或碰撞的极短时间内，摩擦或重力等外力可以忽略，因为内力远大于外力。这种近似使守恒定律在许多实际问题中成立。

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