关于“力-功率等价”的补充说明 （in English and Chinese）

Since the author recently published two blog posts

https://blog.sciencenet.cn/home.php?mod=space&uid=553379&do=blog&id=1295413

and

https://blog.sciencenet.cn/home.php?mod=space&uid=553379&do=blog&id=1345496

It has attracted the attention and discussion of many netizens. One of the key issues is the question of "the equivalence of force and power", that is, physics textbooks generally write the relationship between power and force as

(1)

Indeed, it is an expression, but it is not an expression of "equivalence of force and power". Because the velocity on the right side of equation (1) can take any value, each velocity is related to the following special factors: the force environment, the physical process of the force, the mass of the force object, the resistance encountered, etc.  The equivalence relation pointed out by the author has nothing to do with these factors and is objective and direct.

For example, in an airplane traveling at 300 meters per second, a passenger pushes an object with a force F of 1 newton (equivalent to 1 watt of output power). Viewed from the ground, according to equation (1), the corresponding power is then equal to 300 watts. In fact, this is the result of the help of the plane.

To give another example, when t = 0, a fixed force of 1 Newton (equivalent to a power output of 1 watt) begins to push a primitive stationary object of mass M for uniform acceleration. When t = T, the speed of the object reaches v(T), according to the original physical meaning of formula (1): the work done by the force to the object during this time should be the kinetic energy K(T) obtained by the object,

   （2）

Using the usual relationship between power and kinetic energy, the power P(T)  reaches

   (3)

where a(T) is the acceleration (equal to a constant in the process). It is not difficult to see that equation (3) is equivalent to equation (2). Obviously, the power calculated in this way is related to the physical process. If you choose different time T, the power number P(T) obtained will be different. But the objective relationship, the force F = 1 Newton is equivalent to the power of 1 watt, has never changed!

自从作者最近发布了两篇博文

没有不做功的力--马德堡半球实验、举重运动消耗的卡路里及辐射压公式

和

Force is equivalent to Power

发现引起不少网友的关注与讨论，其中一个焦点问题就是关于“力与功率等价”的疑问，即物理教课书一般都将功率与力的关系写成

         （1）

的确，它是一种表达，但该式不是“力与功率等价”的表达。 因为式(1)右边的速度可以取任意值，而每个速度都与以下特殊因素有关：受力环境、受力的物理过程、受力物体的质量、遇到的阻力等。 笔者所指出的等价关系与这些因素无关，是客观、直接的。

例如，在以每秒 300 米的速度飞行的飞机上，乘客以 1 牛顿的力（相当于 1 瓦的输出功率）推动物体。 从地面看，根据等式（1），相应的功率等于 300 瓦。 其实，这都是飞机帮忙的结果

又如，当 t=0 时，一个1牛顿的固定力 F（相当于1瓦的功率输出）开始推动一个质量为M的原始静止物体进行匀加速，当 t=T 时，物体的速度达到 v(T)，根据式(1)的原始物理意义：在这段时间内力对物体所做的功应该是物体获得的动能 K(T)，

   （2)

使用功率和动能之间的通常关系，功率 P(T) 达到

  (3)

其中 a(T) 是加速度（等于过程中的常数）。 不难看出方程（3）等价于方程（2）。 显然，这种方式计算的功率与物理过程有关。 如果选择不同的时间 T，得到的功率 P(T) 也会不同。 但是客观的关系，力 F=1 牛顿等价于 1 瓦特的功率，始终没有变！