Problem 53 - Physics for Scientists and Engineers with Modern Physics

Question

$$ \begin{array}{l}{\text { A heavy rainstorm dumps } 1.0 \mathrm{cm} \text { of rain on a city } 5 \mathrm{km} \text { wide }} \\ {\text { and } 8 \mathrm{km} \text { long in a } 2 \text { -h period. How many metric tons }} \\\ {\left(1 \text { metric ton }=10^{3} \mathrm{kg}\right) \text { of water fell on the city? }\left(1 \mathrm{cm}^{3} \text { of }\right.} \\ {\text { water has a mass of } 1 \mathrm{g}=10^{-3} \mathrm{kg} . \text { ) How many gallons }} \\ {\text { of water was this? }}\end{array} $$

Step-by-Step Solution

Answer

400,000 metric tons of water fell, equivalent to approximately 105,668,823 gallons.

Step 1: Convert Area to Square Meters

First, we need to calculate the area of the city in square meters. The city is 5 km wide and 8 km long, so its area is \(5 \times 8 = 40\) square kilometers. Since \(1 \text{ km}^2 = 1,000,000 \text{ m}^2\), the area in square meters is \(40 \times 1,000,000 = 40,000,000 \text{ m}^2\).

Step 2: Determine Volume of Rain

The volume of rain that fell is equal to the area of the city multiplied by the depth of the rain. The depth of the rain is 1.0 cm, which needs to be converted to meters: 1.0 cm = 0.01 m. So, the volume is: \(40,000,000 \text{ m}^2 \times 0.01 \text{ m} = 400,000 \text{ m}^3\).

Step 3: Calculate Mass of the Water in Kilograms

Since 1 cubic meter of water has a mass of 1000 kg, the total mass of the water in kilograms is \(400,000 \text{ m}^3 \times 1000 \text{ kg/m}^3 = 400,000,000 \text{ kg}\).

Step 4: Convert Kilograms to Metric Tons

To find the number of metric tons, we divide the mass in kilograms by 1000 (since 1 metric ton = 1000 kg): \(\frac{400,000,000 \text{ kg}}{1000} = 400,000 \text{ metric tons}\).

Step 5: Calculate Gallons of Water

To find the gallons of water, first convert cubic meters to liters. Since one cubic meter is 1000 liters, the volume is \(400,000 \text{ m}^3 \times 1000 = 400,000,000 \text{ liters}\). Then convert liters to gallons, using the conversion factor 1 gallon = 3.78541 liters: \( \frac{400,000,000}{3.78541} \approx 105,668,823 \text{ gallons}\).

Key Concepts

Rainfall Calculation

Rainfall calculation is an important concept in understanding the amount of water that falls in a specific area over a certain period. In this scenario, we want to measure how much rain fell on a city, given the city dimensions, during a heavy storm. We're told the city is 5 km wide and 8 km long, resulting in an area of 40 square kilometers.

To find out the volume of the rain that fell, we multiply the area of the city by the depth of the rainfall. The depth given is 1 cm, but in calculations involving larger scales, it's useful to convert units. Thus, 1 cm is converted to 0.01 meters. This gives us the formula: area multiplied by depth equals volume.

Therefore, the volume of rain is:

• Area of the city in meters: 40 km² x 1,000,000 m²/km² = 40,000,000 m²
• Rain depth in meters: 0.01 m
• Volume of rain: 40,000,000 m² x 0.01 m = 400,000 m³

Understanding these steps helps in analyzing weather data and can be crucial for city planning and flood management.

Metric Conversion

Metric conversion is fundamental in physics and helps us handle measurements across different systems. In this exercise, converting between volume and mass units is critical.

We have calculated the rainfall volume in cubic meters. Now, we need these to be expressed in kilograms to find the mass of the rainwater. Each cubic meter of water is equivalent to 1,000 kilograms because the unit for water density is typically 1,000 kg/m³. Thus, our mass calculation is straightforward:

• Volume: 400,000 m³
• Mass: 400,000 m³ x 1,000 kg/m³ = 400,000,000 kg

Next, we convert the mass from kilograms to metric tons because it’s useful to express large masses in manageable units:
1 metric ton is 1,000 kilograms, so:

• Mass in metric tons: 400,000,000 kg / 1,000 kg/metric ton = 400,000 metric tons

Mastering metric conversions aids in transitioning between units smoothly, essential for science and everyday life.

Mass and Volume

The relationship between mass and volume is key in calculating the impacts of natural events like rainfall. Mass is the amount of matter in an object, while volume is the space that matter occupies. In this context, we’ve determined the volume of water through area multiplication by depth (both converted into compatible units).

Once we have the volume (400,000 m³), determining the mass involves using water’s density (1,000 kg/m³). This tells us how much mass is contained within that volume:

• Mass: Volume x Density = 400,000 m³ x 1,000 kg/m³ = 400,000,000 kg

The next step extends this understanding into the conversion to a more common volumetric unit, gallons. Since 1 m³ equals 1,000 liters and there are roughly 3.78541 liters in a gallon:

• Liter conversion: Volume x 1,000 = 400,000,000 liters
• Gallon conversion: 400,000,000 liters / 3.78541 = approximately 105,668,823 gallons

Understanding these measurements ensures accurate representation of real-world phenomena and supports practical applications such as water resource management.