---
title: "Water depth after ice melts"
subject: "Fluid Mechanics"
chapter: 3
key: "semester_3/fluid_mechanics:ice-melting-water-depth"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/ice-melting-water-depth
site: BCT Engineering
---

# Water depth after ice melts

Fluid Mechanics · Chapter 3: Hydrostatics

- Asked in: 2075 Chaitra, 8 marks
- Video: [Does Melting Ice Raise the Water Level? | Physics with Professor Matt Anderson | M15-04](https://www.youtube.com/watch?v=lT8WgGQzOHE) — Physics with Professor Matt Anderson
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/ice-melting-water-depth
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-3

## Question

0.5 m³ of ice floats in a cylindrical tank maintaining 4 m depth as shown in figure below. What will be the depth of water if ice completely melt in the tank? [Figure: cylindrical tank of water 4 m deep; ice block of total volume 0.5 m³ floating, with 0.2 m³ above the water level and 0.3 m³ below]

## Short answer

The melted water occupies exactly the displaced volume (0.3 m³), so the total liquid volume below the original surface does not change. The water depth remains 4 m.

## Answer

**Answer: the depth remains 4 m; the water level does not change.**

**Reasoning (Archimedes' principle).** Take water density $\rho$. The floating ice displaces a volume $V_d=0.3\ \text{m}^3$ (the part below the surface). By equilibrium, the weight of the ice equals the weight of water displaced:

$$
W_{ice}=\rho g V_d=\rho g(0.3)
$$

So the ice has a mass of $0.3\rho$. When it melts, it turns into water of the same mass, with volume

$$
V_{melt}=\frac{W_{ice}}{\rho g}=0.3\ \text{m}^3
$$

The melted water occupies exactly the 0.3 m³ that the submerged part of the ice used to displace, so the total volume of liquid below the original surface is unchanged. The 0.2 m³ of ice above the surface was already part of the ice's weight and is included in the 0.3 m³ of melt water.

$$
h_{new}=4\ \text{m}
$$

This holds because the ice floats in water of the same kind (pure water). If fresh-water ice melted in denser sea water, the level would rise slightly.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. When the ice melts, what happens to the water depth?
   A. Stays at 4 m
   B. Rises by 0.2 m^3 worth of depth
   C. Falls by 0.2 m^3 worth of depth
   D. Rises by 0.5 m^3 worth of depth
   Answer: A. Melt water exactly fills the volume the submerged ice displaced. (Key point: Answer)

2. What does the floating ice satisfy?
   A. Its weight equals the weight of water displaced (0.3 m^3)
   B. Its weight equals the weight of 0.5 m^3 of water
   C. Its weight equals the weight of 0.2 m^3 of water
   D. Its volume equals the displaced volume
   Answer: A. Archimedes: W_ice = ρ g V_d with V_d = 0.3 m^3 (the submerged part). (Key point: Floating equilibrium)

3. What volume of water does the melted ice form?
   A. 0.5 m^3
   B. 0.3 m^3, equal to the displaced volume
   C. 0.2 m^3
   D. 0.8 m^3
   Answer: B. The melted mass is 0.3 ρ, so its volume at density ρ is 0.3 m^3. (Key point: Melt volume)

4. If fresh-water ice melted in denser sea water, what would happen to the level?
   A. It would fall
   B. It would stay the same
   C. It would rise slightly
   D. It would double
   Answer: C. In denser water the ice displaces less than its melt volume, so melt water adds extra volume. (Key point: Sea water case)

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