Problem 53 - Physics for Scientists and Engineers with Modern Physics

Question

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

Step-by-Step Solution

Answer

400,000 metric tons, approximately 105,668,821 gallons.

Step 1: Calculate the Volume of Rainfall

To find the volume of rainfall, multiply the area of the city by the depth of the rainfall. The area of the city is given as 5 km by 8 km. First, convert kilometers to centimeters: - 1 km = 100,000 cm, so 5 km = 500,000 cm and 8 km = 800,000 cm. Now calculate the area: \[\text{Area} = 500,000 \text{ cm} \times 800,000 \text{ cm} = 400,000,000,000 \text{ cm}^2. \]The depth of the rain is 1 cm, so the volume \(V\) is:\[V = \text{Area} \times \text{Depth} = 400,000,000,000 \text{ cm}^2 \times 1 \text{ cm} = 400,000,000,000 \text{ cm}^3.\]

Step 2: Calculate Mass of the Water in Kilograms

Convert the volume of water to mass since 1 cm³ of water has a mass of 1 gram:\[\text{Mass} = 400,000,000,000 \text{ cm}^3 \times 1 \text{ g/cm}^3 = 400,000,000,000 \text{ g}.\]Convert grams to kilograms since 1 kg = 1,000 g:\[\text{Mass in kg} = \frac{400,000,000,000 \text{ g}}{1,000} = 400,000,000 \text{ kg}.\]

Step 3: Convert Mass to Metric Tons

We know that 1 metric ton = 1,000 kg, therefore:\[\text{Mass in metric tons} = \frac{400,000,000 \text{ kg}}{1,000} = 400,000 \text{ metric tons}.\]

Step 4: Convert Mass to Gallons

To convert kilograms of water to gallons, use the conversion factor that 1 gallon is approximately 3.78541 liters. Knowing that 1 liter of water has a mass of 1 kg:First, find the volume in liters:\[400,000,000 \text{ kg} = 400,000,000 \text{ liters}.\]Now convert liters to gallons:\[\text{Volume in gallons} = \frac{400,000,000 \text{ liters}}{3.78541} \approx 105,668,821 \text{ gallons}.\]

Key Concepts

Rainfall Measurement

Measuring rainfall is essential in understanding the amount of water that reaches the ground during storms. Rainfall can be measured in terms of depth, for example, in centimeters or inches. This measurement reflects how much water would collect on a flat surface without any drainage. It's important to convert this depth to a volume to understand the total water collected.

To calculate the volume from rainfall depth, determine the area it falls over. As in the given exercise, convert all measurements into the same unit, such as centimeters. Then, multiply the area by the rainfall depth.

• The area in our example is calculated by converting the city dimensions from kilometers to centimeters.
• Next, multiplying this area by the 1 cm depth of rain gives the volume.

No matter the scale, whether it's a small garden or an entire city, the process remains the same.

Unit Conversion

Unit conversion is a vital skill in physics and everyday calculations. It allows us to switch between different systems of measurement, ensuring our calculations and results are meaningful. In the context of the exercise, several unit conversions are pivotal to solve the problem.

• First, converting kilometers to centimeters simplifies the area calculation for rainfall measurement. Since 1 km = 100,000 cm, convert both the width and length of the city appropriately.
• Next, converting mass from grams to kilograms is necessary since 1 cm³ of water's mass is 1 gram. Hence, divide by 1,000 to get kilograms.
• Finally, for the mass, convert kilograms into metric tons, where 1 metric ton equals 1,000 kg.

Understanding these fundamental conversions can greatly streamline problem-solving, enabling accurate and efficient solutions.

Metric System

The metric system is a standardized system of measurement used globally, especially in science and industry. It's based on units that scale by powers of ten, providing an intuitive way to convert between different units.

In the example problem, we use metric units for measuring rain (centimeters, grams, kilograms, and metric tons). This makes calculations straightforward because each unit step relies on a base-10 system.

• Distance is first measured in kilometers, which is converted into centimeters for precise area calculation.
• Water volume is initially determined in cubic centimeters for ease with metric mass units (grams).
• Conversions among metric units are intuitive and consistent, like grams to kilograms or kilograms to metric tons.

Understanding and utilizing the metric system effectively is instrumental for calculations concerning measurement and conversions, leading to less error and more clarity in problem-solving.