Momentum | 动量

📚 Momentum | 动量

Momentum is one of the most important quantities in mechanics. It links mass, velocity, force and time, and it is conserved in all collisions and explosions when no external resultant force is acting. A clear understanding of momentum is essential for solving a wide range of CIE A-Level Physics problems, from one-dimensional collisions to two-dimensional recoil events.

动量是力学中最重要的物理量之一。它把质量、速度、力和时间联系起来，并且当没有外合力作用时，动量在所有的碰撞和爆炸过程中守恒。清晰理解动量对于解决 CIE A-Level 物理中从一维碰撞到二维反冲事件的各类问题都至关重要。

1. Defining Momentum | 定义动量

Linear momentum p of an object is defined as the product of its mass m and its velocity v. In symbols, momentum is written as p = m v.

物体的线动量 p 定义为其质量 m 与速度 v 的乘积。用符号表示，动量写作 p = m v。

p = m v

Momentum is measured in kg m s⁻¹. Since a newton is equal to kg m s⁻², the unit kg m s⁻¹ is equivalent to N s.

动量的单位是 kg m s⁻¹。由于牛顿等于 kg m s⁻²，所以单位 kg m s⁻¹ 等价于 N s。

For example, a 2.0 kg object moving at 3.0 m s⁻¹ has a momentum of p = 2.0 × 3.0 = 6.0 kg m s⁻¹.

例如，一个 2.0 kg 的物体以 3.0 m s⁻¹ 的速度运动，其动量为 p = 2.0 × 3.0 = 6.0 kg m s⁻¹。

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2. Vector Nature of Momentum | 动量的矢量性

Because velocity is a vector, momentum is also a vector. The direction of momentum is always the same as the direction of the object’s velocity.

因为速度是矢量，动量也是矢量。动量的方向始终与物体速度的方向相同。

In one-dimensional problems, you must choose a positive direction. A momentum to the right may be taken as positive, while a momentum to the left is negative.

在一维问题中，必须选择一个正方向。向右的动量可以取为正，而向左的动量为负。

For instance, if a 2.0 kg object moves right at 3.0 m s⁻¹, its momentum is +6.0 kg m s⁻¹. If the same object moves left at 3.0 m s⁻¹, its momentum is -6.0 kg m s⁻¹.

例如，如果一个 2.0 kg 的物体向右以 3.0 m s⁻¹ 运动，其动量为 +6.0 kg m s⁻¹。如果同一物体向左以 3.0 m s⁻¹ 运动，其动量为 -6.0 kg m s⁻¹。

Treating momentum as a vector is essential when adding momenta in collisions or explosions.

在处理碰撞或爆炸中的动量求和时，将动量视为矢量是必不可少的。

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

Newton’s second law can be expressed in terms of momentum: the resultant force on an object is equal to the rate of change of its momentum.

牛顿第二定律可以用动量来表述：物体所受的合力等于其动量的变化率。

F = Δp / Δt

If the mass of the object is constant, then Δp = m Δv, so the equation becomes F = m Δv / Δt = m a. This is the familiar form F = ma.

如果物体的质量不变，则 Δp = m Δv，因此方程变为 F = m Δv / Δt = m a。这就是熟悉的形式 F = ma。

The momentum form is more general than F = ma because it can also be applied when mass changes, such as in rocket motion where fuel is ejected.

动量形式比 F = ma 更普遍，因为它也适用于质量变化的情况，例如燃料被喷出的火箭运动。

In CIE examinations, the formula F = Δp / Δt is often used to find the average force acting during a collision.

在 CIE 考试中，公式 F = Δp / Δt 常被用来求碰撞过程中作用的平均力。

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4. Impulse and the Impulse-Momentum Theorem | 冲量与冲量-动量定理

Impulse J is defined as the product of a constant resultant force F and the time interval Δt for which it acts. In symbols, J = F Δt.

冲量 J 定义为恒定合力 F 与其作用时间间隔 Δt 的乘积。用符号表示为 J = F Δt。

J = F Δt

The impulse-momentum theorem states that the impulse acting on an object is equal to the change in momentum of that object: J = Δp.

冲量-动量定理指出，作用在物体上的冲量等于该物体动量的变化：J = Δp。

Impulse is a vector quantity. Its direction is the same as the direction of the applied force or the change in momentum. The SI unit of impulse is N s, which is equivalent to kg m s⁻¹.

冲量是一个矢量。它的方向与作用力或动量变化的方向相同。冲量的国际单位是 N s，等价于 kg m s⁻¹。

For example, if a force of 50 N acts on a stationary 10 kg cart for 4.0 s, the impulse is 50 × 4.0 = 200 N s. Since the initial momentum is zero, the final momentum is 200 kg m s⁻¹.

例如，如果 50 N 的力作用在静止的 10 kg 小车上 4.0 s，冲量为 50 × 4.0 = 200 N s。由于初始动量为零，最终动量为 200 kg m s⁻¹。

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5. Force-Time Graphs | 力-时间图像

When a force is not constant, the impulse can be found from the area under a force-time graph.

当力不是恒力时，冲量可以通过力-时间图像下方的面积求得。

Since J = F Δt for a constant force, a rectangle of height F and width Δt gives the impulse. For a varying force, you must find the area by counting squares, using geometry or integrating if the function is known.

由于恒力情况下 J = F Δt，因此高度为 F、宽度为 Δt 的矩形面积就表示冲量。对于变力，必须通过数格、几何方法或在函数已知时积分来求面积。

The area under a force-time graph between two times equals the change in momentum of the object over that time interval.

两个时间之间力-时间图像下方的面积等于物体在该时间间隔内动量的变化。

A typical exam question gives a triangular or trapezoidal force-time graph and asks for the final speed of an object. Calculate the area first to obtain Δp, then divide by mass to find Δv.

典型的考题给出一幅三角形或梯形的力-时间图像，并要求求物体的末速度。先计算面积得到 Δp，再除以质量求 Δv。

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6. Conservation of Momentum | 动量守恒

The principle of conservation of momentum states that when no external resultant force acts on a system, the total momentum of the system remains constant.

动量守恒定律指出，当系统不受外合力作用时，系统的总动量保持不变。

For two objects colliding, total momentum before the collision equals total momentum after the collision:

对于两个碰撞物体，碰撞前的总动量等于碰撞后的总动量：

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Here u represents initial velocity and v represents final velocity. The subscripts 1 and 2 label the two objects.

这里 u 表示初速度，v 表示末速度。下标 1 和 2 分别标记两个物体。

Conservation of momentum follows from Newton’s third law. During a collision, the forces between two objects are equal in magnitude and opposite in direction, and they act for the same time. Therefore the impulses on each object are equal and opposite, so the total change in momentum of the system is zero.

动量守恒来自牛顿第三定律。碰撞过程中，两个物体之间的力大小相等、方向相反，并且作用时间相同。因此每个物体受到的冲量等大反向，系统总动量的变化为零。

In real collisions, external forces such as friction are often small during the very short collision time, so momentum conservation can still be applied as a very good approximation.

在真实碰撞中，由于碰撞时间极短，摩擦力等外力通常很小，所以动量守恒仍然可以作为很好的近似应用。

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7. Elastic and Inelastic Collisions | 弹性碰撞与非弹性碰撞

An elastic collision is one in which both momentum and total kinetic energy are conserved. In macroscopic collisions, truly elastic collisions are rare, but collisions between gas molecules are often treated as elastic.

弹性碰撞是动量和总动能都守恒的碰撞。在宏观碰撞中，真正的弹性碰撞很少，但气体分子之间的碰撞通常被视为弹性碰撞。

An inelastic collision conserves momentum but does not conserve total kinetic energy. Some kinetic energy is transformed into heat, sound or permanent deformation.

非弹性碰撞动量守恒但总动能不守恒。部分动能转化为热能、声能或永久形变。

A perfectly inelastic collision is a special case in which the objects stick together after impact and move with a common velocity.

完全非弹性碰撞是一种特殊情况，碰撞后物体粘在一起并以共同速度运动。

Type of collision | 碰撞类型	Momentum | 动量	Kinetic energy | 动能	Feature | 特征
Elastic | 弹性	Conserved | 守恒	Conserved | 守恒	Objects bounce apart | 物体弹开
Inelastic | 非弹性	Conserved | 守恒	Not conserved | 不守恒	Some KE is lost | 部分动能损失
Perfectly inelastic | 完全非弹性	Conserved | 守恒	Not conserved | 不守恒	Objects stick together | 物体粘在一起

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8. Solving One-Dimensional Collision Problems | 一维碰撞问题求解

For one-dimensional collision problems, start by choosing a positive direction and writing down the given velocities with the correct signs.

对于一维碰撞问题，首先要选择正方向，并用正确的符号写出已知速度。

Apply the conservation of momentum equation: total momentum before = total momentum after.

应用动量守恒方程：碰撞前总动量 = 碰撞后总动量。

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

If the objects stick together, the collision is perfectly inelastic. Set v₁ = v₂ = v, giving:

如果物体粘在一起，碰撞为完全非弹性碰撞。令 v₁ = v₂ = v，得到：

(m₁ + m₂)v = m₁u₁ + m₂u₂

If the collision is elastic, you must also use conservation of kinetic energy:

如果碰撞是弹性碰撞，还必须使用动能守恒：

½m₁u₁² + ½m₂u₂² = ½m₁v₁² + ½m₂v₂²

Worked example: a 2.0 kg trolley moving at 3.0 m s⁻¹ catches up with a 1.0 kg trolley moving at 1.0 m s⁻¹. They stick together on impact. By conservation of momentum, 2.0 × 3.0 + 1.0 × 1.0 = 3.0 × v, so 7.0 = 3.0v and v = 2.33 m s⁻¹.

例题：一辆 2.0 kg 的小车以 3.0 m s⁻¹ 追上一辆以 1.0 m s⁻¹ 运动的 1.0 kg 小车。它们碰撞后粘在一起。根据动量守恒，2.0 × 3.0 + 1.0 × 1.0 = 3.0 × v，所以 7.0 = 3.0v，v = 2.33 m s⁻¹。

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9. Explosions and Recoil | 爆炸与反冲

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