泳池里的船

floating in a swimming pool

专题: Brainteaser / 脑筋急转弯
难度: L2
来源: QuantQuestion

个人笔记

题目详情

1）一艘小船漂在游泳池里。船上有一块体积很小但非常重的石头。如果把石头从船上扔进泳池里，泳池水位会发生什么变化？

2）你坐在泳池里的小船上，把船锚扔到水里。泳池水位会发生什么变化？

英文原题

A small boat is floating in a swimming pool. The boat contains a very small but very heavy rock. If the rock is tossed out of the boat into the pool, what happens to the water level in the pool?
You are sitting in a boat in a swimming pool, you throw the anchor over the side. What happens to the level of the water in the pool?

解析

关键：物体在船上时，船为了额外承重，需要多排开与该重量等重的水；物体落入水中后，若其沉底，则只排开自身的体积。

(1) 扔石头：

石头在船上时，额外排水体积为 $\Delta V_\text{boat}=\frac{W}{\rho_w g}$ （与重量成正比）。
石头扔进水里并沉底时，额外排水体积为石头体积 $\Delta V_\text{rock}=\frac{W}{\rho_\text{rock} g}$ 。

由于石头密度 $\rho_\text{rock}>\rho_w$ ，所以 $\Delta V_\text{rock}<\Delta V_\text{boat}$ ，总排水体积变小，因此水位下降。

(2) 扔锚：通常锚的密度也大于水、会下沉，同理：

在船上时按“重量”排水；
扔进水里后按“体积”排水；

因 $\rho_\text{anchor}>\rho_w$ ，排水减少，水位也下降。

英文解析

Key: When an object is on the boat, the boat displaces an additional volume of water equal in weight to the object; when the object falls into the water and sinks, it displaces only its own volume.

(1) Throwing a stone:

When the stone is on the boat, the additional displaced volume is $\Delta V_\text{boat}=\frac{W}{\rho_w g}$ (proportional to weight).
When the stone is thrown into the water and sinks, the additional displaced volume is the stone's volume $\Delta V_\text{rock}=\frac{W}{\rho_\text{rock} g}$ .

Since the stone's density $\rho_\text{rock}>\rho_w$ , it follows that $\Delta V_\text{rock}<\Delta V_\text{boat}$ . The total displaced volume decreases, so the water level falls.

(2) Throwing an anchor: An anchor typically has a density greater than water and will sink; similarly:

It displaces water based on its "weight" while on the boat;
It displaces water based on its "volume" after being thrown into the water.

Since $\rho_\text{anchor}>\rho_w$ , the displaced volume decreases, and the water level also falls.