Problem 73 - Chemistry The Central Science In Si Units

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 \(350 \mathrm{~km} /\) 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) 386,241,600 meters, (b) ~3,973,964 seconds, (c) ~2.574 seconds, (d) ~66,636 mi/hr.

Step 1: Convert distances from miles to meters

To convert the distance from miles to meters, use the conversion factor: \[ 1 ext{ mile} = 1609.34 ext{ meters} \]Given the distance in miles is 240,000, the conversion to meters is:\[ 240,000 ext{ miles} \times 1609.34 \frac{m}{ ext{mile}} = 3.862416 imes 10^8 ext{ meters} \]

Step 2: Convert falcon's speed to meters per second

First convert the falcon's speed from km/hr to m/s. Use the conversion factors:\[ 1 ext{ km} = 1000 ext{ m} \text{ and } 1 ext{ hour} = 3600 ext{ seconds} \]\[ 350 ext{ km/hr} = 350 \times 1000 ext{ m/hr} = 350,000 ext{ m/hr} \]Then convert to m/s:\[ 350,000 ext{ m/hr} \div 3600 ext{ s/hr} = 97.222 ext{ m/s} \]

Step 3: Calculate time for the falcon to reach the Moon

To find the time in seconds, divide the distance to the Moon by the falcon's speed in m/s:\[ \text{Time} = \frac{3.862416 \times 10^8 ext{ meters}}{97.222 ext{ m/s}} \approx 3,973,964 ext{ seconds} \]

Step 4: Calculate time for light to travel to the Moon and back

First, find the one-way trip time:\[ \text{Time to Moon} = \frac{3.862416 \times 10^8 ext{ meters}}{3 \times 10^8 ext{ m/s}} \approx 1.287 ext{ seconds} \]Round trip time would be twice this:\[ 1.287 \times 2 = 2.574 ext{ seconds} \]

Step 5: Convert Earth's speed from km/s to mi/hr

Convert 29.783 km/s to mi/hr using the conversion factors:\[ 1 ext{ km} = 0.621371 ext{ miles} \text{ and } 1 ext{ hour} = 3600 ext{ seconds} \]\[ 29.783 ext{ km/s} \times 0.621371 ext{ mi/km} = 18.510 ext{ mi/s} \]Then convert to mi/hr:\[ 18.510 ext{ mi/s} \times 3600 ext{ s/hr} = 66,636 ext{ mi/hr} \]

Key Concepts

Distance Calculation

Understanding how to calculate distances is crucial in physics and astronomy. When you have a distance given in one unit, like miles, you might need to convert it to a different unit, such as meters, especially in scientific contexts.
Using the conversion factor, where 1 mile equals 1609.34 meters, you can easily perform this calculation. For a distance of 240,000 miles to the Moon, the converted distance becomes:

• Distance in meters = 240,000 miles × 1609.34 meters/mile
• = 3.862416 × 10⁸ meters

Breaking this process into steps helps ensure that you can handle different units effortlessly. Practicing conversions strengthens your understanding of different distance measures.

Speed Calculation

Speed calculations are a fun way to apply math to real-world scenarios, like figuring out how quickly a falcon could reach the Moon. Speed is often given in various units, and converting between them is a key skill.
For the peregrine falcon, the speed was initially given in km/hr and needed converting to m/s using two steps. First, convert km to meters, then change the hour to seconds:

• 350 km/hr = 350,000 meters/hr
• In m/s = 350,000 m/hr ÷ 3600 seconds/hr
• ≈ 97.222 m/s

This conversion allows you to directly compare and compute speeds, which is especially valuable when integrating calculations into broader physics problems. Knowing these conversions boosts your confidence in tackling complex problems.

Light Speed

Light speed underpins many fascinating physics phenomena, including how long it takes light to travel vast distances. Light moves at approximately 3.00 × 10⁸ m/s, which allows us to calculate the time it takes for light to travel to and from different celestial bodies.
To find how long it takes light to make a round trip to the Moon:

• Time for a one-way trip = (distance in meters) ÷ (speed of light)
• = 3.862416 × 10⁸ meters ÷ 3.00 × 10⁸ m/s
• ≈ 1.287 seconds each way
• Round trip = 1.287 seconds × 2
• ≈ 2.574 seconds

Understanding these calculations not only adds to your scientific knowledge but also enhances your appreciation of how quickly information, in the form of light, can traverse the cosmos.

Metric Conversions

Converting between metric units and other measurement systems like miles to kilometers, or vice versa, is an essential skill. Whether in physics exercises or real-world applications, these conversions ensure you're working with consistent units.
Let's take the speed of Earth, initially given in km/s, and convert it to mi/hr. With the following steps:

• Convert km to miles using 1 km = 0.621371 miles
• 29.783 km/s × 0.621371 mi/km = 18.510 mi/s
• To convert mi/s to mi/hr, multiply by 3600 seconds/hr
• 18.510 mi/s × 3600 s/hr = 66,636 mi/hr

This ensures your data is in a familiar format, making understanding and communication of scientific information easier. Mastering these conversions not only helps in academic settings but also in making informed decisions in everyday life.