G.P.E.–K.E. Transformations | 重力势能与动能转化

📚 G.P.E.–K.E. Transformations | 重力势能与动能转化

Gravitational potential energy (g.p.e.) and kinetic energy (k.e.) are two of the most important mechanical energy stores in Cambridge International A-Level Physics. When an object moves under gravity and resistive forces are negligible, energy is continually transformed between these two forms. Understanding these transformations allows you to solve motion problems using energy conservation rather than kinematics.

重力势能（g.p.e.）和动能（k.e.）是剑桥国际 A-Level 物理中最重要的两种机械能储存形式。当物体在重力作用下运动且阻力可忽略时，能量会在这两种形式之间不断转换。理解这些转化使你能够用能量守恒而非运动学方程来解决运动问题。

1. Energy in a Gravitational Field | 重力场中的能量

Gravitational potential energy is energy stored by an object because of its position in a gravitational field. Near the Earth’s surface, the change in g.p.e. is given by ΔEₚ = mgΔh, where m is mass, g is the gravitational field strength, and Δh is the vertical height change. Only changes in g.p.e. are physically meaningful; the zero level can be chosen anywhere for convenience.

重力势能是物体因在重力场中的位置而储存的能量。在地球表面附近，重力势能的变化量为 ΔEₚ = mgΔh，其中 m 为质量，g 为重力场强度，Δh 为竖直高度变化。只有重力势能的变化量才有物理意义；零势能面可以根据方便任意选取。

Near the Earth’s surface, g is approximately 9.81 m s⁻², which is equivalent to 9.81 N kg⁻¹. In Cambridge exams, the value of g is usually given as 9.81 m s⁻², but questions sometimes specify 10 m s⁻² for simpler calculation.

在地球表面附近，g 约为 9.81 m s⁻²，等价于 9.81 N kg⁻¹。在剑桥考试中，g 通常取 9.81 m s⁻²，但题目有时会指定 10 m s⁻² 以简化计算。

2. Kinetic Energy | 动能

Kinetic energy is the energy an object possesses because of its motion. For an object of mass m moving at speed v, the kinetic energy is Eₖ = ½mv². Because speed is squared, kinetic energy increases rapidly as speed increases, and kinetic energy is always a positive scalar quantity.

动能是物体因运动而具有的能量。对于质量为 m、速度为 v 的物体，动能为 Eₖ = ½mv²。由于速度被平方，动能随速度增大而迅速增加，且动能始终为正的标量。

In g.p.e.–k.e. transformations, a falling object converts gravitational potential energy into kinetic energy. Conversely, an object moving upward converts kinetic energy back into gravitational potential energy. These two processes are reversible in an ideal system with no resistive forces.

在重力势能与动能转化中，下落物体将重力势能转化为动能。相反，上升物体将动能重新转化为重力势能。在无阻力的理想系统中，这两个过程是可逆的。

3. Mechanical Energy Conservation | 机械能守恒

If gravity is the only force doing work, or if non-conservative forces such as air resistance and friction are negligible, the total mechanical energy of the system remains constant: Eₚ + Eₖ = constant. This principle is often easier to apply than equations of motion because it relates speed directly to height change.

如果做功的力只有重力，或者空气阻力、摩擦力等非保守力可以忽略，则系统总机械能保持不变：Eₚ + Eₖ = 常数。这一原理通常比运动学方程更易应用，因为它直接将速度与高度变化联系起来。

The condition ‘no air resistance, no friction’ is essential. When these forces are present, they do negative work and convert some mechanical energy into internal energy, so the total mechanical energy decreases. Exam questions frequently ask you to identify this energy loss.

“无空气阻力、无摩擦”这一条件至关重要。当这些力存在时，它们做负功，将部分机械能转化为内能，因此总机械能减少。考试题目经常要求你识别这种能量损失。

Eₚ + Eₖ = constant

4. Free Fall: From Rest | 自由落体：从静止开始

Consider an object released from rest at a height h above a chosen reference level. Initially, its g.p.e. is mgh and its kinetic energy is zero. Just before impact, its g.p.e. is zero and its kinetic energy is ½mv². Conservation of mechanical energy gives mgh = ½mv².

考虑一个物体从参考面以上高度 h 处由静止释放。初始时，其重力势能为 mgh，动能为零。在即将撞击前，其重力势能为零，动能为 ½mv²。机械能守恒给出 mgh = ½mv²。

mgh = ½mv² ⇒ v = √(2gh)

Mass cancels from the equation, so the final speed is independent of mass. This explains why, in a vacuum, a feather and a hammer fall at the same rate and reach the ground with the same speed for the same height.

质量从方程中约去，因此末速度与质量无关。这解释了为什么在真空中，羽毛和铁锤以相同速率下落，并在相同高度处达到相同末速度。

If the object has an initial downward speed u, the energy equation becomes mgh + ½mu² = ½mv². Rearranging gives v² = u² + 2gh, which is identical to the corresponding equation of uniformly accelerated motion.

若物体有向下的初速度 u，能量方程变为 mgh + ½mu² = ½mv²。整理可得 v² = u² + 2gh，这与匀加速运动的对应方程完全一致。

5. Vertical Projection | 竖直上抛

When an object is thrown vertically upward with initial speed u, it begins with only kinetic energy: Eₖ = ½mu². At the maximum height H, its vertical velocity

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