Problem 67 - Chemistry: The Central Science

Question

The distance from Earth to the Moon is approximately \(240,000 \mathrm{mi}\). (a) What is this distance in meters? (b) The peregrine falcon has been measured as traveling up to \(350 \mathrm{~km} /\) \(\mathrm{hr}\) in a dive. If this falcon could fly to the Moon at this speed, how many seconds would it take? (c) The speed of light is \(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\). How long does it take for light to travel from Earth to the Moon and back again? (d) Earth travels around the Sun at an average speed of \(29.783 \mathrm{~km} / \mathrm{s}\). Convert this speed to miles per hour.

Step-by-Step Solution

Answer

(a) The distance from Earth to the Moon in meters is \(386,241,600 \mathrm{m}\). (b) The peregrine falcon would take approximately \(3,970,357 \mathrm{s}\) to fly to the Moon at its maximum speed. (c) It takes light approximately \(2.574 \mathrm{s}\) to travel from Earth to the Moon and back again. (d) Earth travels around the Sun at an average speed of approximately \(66,629 \frac{\mathrm{mi}}{\mathrm{h}}\).

Step 1: Conversion factor from miles to meters

Use the conversion factor that 1 mile is equal to 1,609.34 meters.

Step 2: Convert distance to meters

Multiply the given distance from Earth to the Moon in miles (240,000 mi) by the conversion factor to get the distance in meters: \(240,000 \times 1,609.34 = 386,241,600 \mathrm{m}\). (b) Calculate time taken by the peregrine falcon to reach the Moon at its maximum speed:

Step 1: Convert speed to meters per second

Convert the peregrine falcon's speed from km/h to m/s: \(350 \frac{\mathrm{km}}{\mathrm{h}} \times \frac{1,000 \mathrm{m}}{1 \mathrm{km}} \times \frac{1 \mathrm{h}}{3,600 \mathrm{s}} \approx 97.22 \frac{\mathrm{m}}{\mathrm{s}}\).

Step 2: Calculate time taken to reach the Moon

Use the formula for calculating time: \(\mathrm{Time} = \frac{\mathrm{Distance}}{\mathrm{Speed}}\). Time taken by the falcon to reach the Moon: \( \frac{386,241,600 \mathrm{m}}{97.22 \frac{\mathrm{m}}{\mathrm{s}}} \approx 3,970,357 \mathrm{s}\). (c) Calculate time taken for light to travel from Earth to the Moon and back again:

Step 1: Calculate time taken for a one-way trip

Use the formula for calculating time: \(\mathrm{Time} = \frac{\mathrm{Distance}}{\mathrm{Speed}}\). Time taken by light for a one-way trip: \(\frac{386,241,600 \mathrm{m}}{3.00 \times 10^8 \frac{\mathrm{m}}{\mathrm{s}}} \approx 1.287 \mathrm{s}\).

Step 2: Calculate time taken for a round-trip

Multiply the time for a one-way trip by 2 to get the time taken for a round trip: \(1.287 \mathrm{s} \times 2 = 2.574 \mathrm{s}\). (d) Convert the Earth's average speed around the Sun from km/s to miles per hour:

Step 1: Conversion factor from km/s to mi/h

Use the following conversion factors: \(1 \mathrm{km} = 0.621371 \mathrm{mi}\) and \(1 \mathrm{s} = \frac{1}{3600} \mathrm{h}\).

Step 2: Convert speed to miles per hour

Convert Earth's speed around the Sun to miles per hour: \(29.783 \frac{\mathrm{km}}{\mathrm{s}} \times \frac{0.621371 \mathrm{mi}}{1 \mathrm{km}} \times \frac{3600 \mathrm{h}}{1 \mathrm{s}} \approx 66,629 \frac{\mathrm{mi}}{\mathrm{h}}\).

Key Concepts

Distance Measurement Conversion

Understanding unit conversion is essential in the field of chemistry, where measurements are a fundamental part of experiments and calculations. One requires to grasp how to convert distance measurements from one unit to another to accurately represent quantities. For instance, in the exercise, the distance from Earth to the Moon is given in miles and needs to be converted to meters because scientific measurements commonly use the metric system.

Starting with the conversion factor that 1 mile is approximately equal to 1,609.34 meters, we multiply the distance in miles by this factor to convert it to meters. This practice is common in scientific disciplines since it aligns with international standards, ensuring seamless communication and consistency in data.

Speed of Light Calculation

The speed of light, denoted as c, is a fundamental constant in physics and is crucial for understanding a variety of astronomical and physical phenomena. In the given problem, calculating how long it takes for light to travel from Earth to the Moon is a practical application of this constant, which is approximately 299,792,458 meters per second, though we often use the rounded value of 3.00 x 10^8 m/s for simplicity.

Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \), we can calculate the time in seconds for light to cover a certain distance. This calculation is not only fascinating but also vital for various technologies such as GPS and for understanding the vast scales of space in astronomy.

Time Calculation for Travel

Calculating the time it takes for travel at a certain speed over a set distance is a skill used in various scientific and real-world contexts. For example, if we want to know how long it will take for a peregrine falcon, the fastest bird, to reach the Moon, we first need to convert its diving speed from kilometers per hour to meters per second, the standard unit for velocity in physics.

Once we convert the speed, we then divide the total distance to the Moon in meters by the falcon’s speed in meters per second, using the aforementioned formula. This calculation is theoretical, demonstrating the potential of using consistent scientific units to make even the most curious computations.

Velocity Conversion

Velocity conversion is an essential process to compare speeds that are given in different units, potentially across different measurement systems. In the exercise, the Earth's average speed around the Sun is provided in kilometers per second, but for a relatable understanding, converting it to miles per hour, a unit familiar in daily life, especially in the United States, is required.

Applying the relevant conversion factors facilitates understanding of the Earth’s movement in terms more commonly used in various sectors, like automotive and aviation. The method consists of multiplying by the factor that converts kilometers to miles and then by the number of seconds in an hour to convert seconds to hours. This practice exemplifies how scientific knowledge is translated into units that connect better with everyday experiences.