Question

Carry out the following conversions: (a) 32.4 yd to centimeters, (b) \(3.0 \times 10^{10} \mathrm{~cm} / \mathrm{s}\) to \(\mathrm{ft} / \mathrm{s},\) (c) 1.42 light-years to miles (a light-year is an astronomical measure of distance \(-\) the distance traveled by light in a year, or 365 days; the speed of light is \(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\) ).

Short answer

(a) 2964.656 cm, (b) \(9.8425 \times 10^{8} \text{ ft/s}\), (c) \(8.34564 \times 10^{12} \text{ miles}\)

Step-by-step solution

Convert Yards to Centimeters

To convert 32.4 yards to centimeters, first convert yards to meters using the conversion factor 1 yard = 0.9144 meters. Then convert meters to centimeters, knowing that 1 meter = 100 centimeters.1. Convert 32.4 yards to meters:\[32.4 \, \text{yd} \times 0.9144 \, \text{m/yd} = 29.64656 \, \text{m}\]2. Convert meters to centimeters:\[29.64656 \, \text{m} \times 100 \, \text{cm/m} = 2964.656 \, \text{cm}\]

Convert Speed from cm/sec to ft/sec

To convert the speed \(3.0 \times 10^{10} \text{ cm/s}\) to feet per second, use the conversion factor 1 foot = 30.48 centimeters.Convert centimeters to feet:\[3.0 \times 10^{10} \, \text{cm/s} \times \left( \frac{1 \, \text{ft}}{30.48 \, \text{cm}} \right) \approx 9.8425 \times 10^{8} \, \text{ft/s}\]

Convert Light-Years to Miles

1. First, calculate the distance light travels in meters in one year. Use the speed of light \(3.00 \times 10^{8} \, \text{m/s}\) and the number of seconds in a year \(365 \times 24 \times 3600\). Distance in meters:\[3.00 \times 10^{8} \, \text{m/s} \times 365 \times 24 \times 3600 \, \text{s} = 9.4608 \times 10^{15} \, \text{meters}\]2. Convert meters to miles using the conversion factor 1 mile = 1609.34 meters.Distance in miles:\[9.4608 \times 10^{15} \, \text{m} \times \left( \frac{1 \, \text{mile}}{1609.34 \, \text{m}} \right) \approx 5.87863 \times 10^{12} \, \text{miles}\]3. Multiply by 1.42 light-years:\[1.42 \times 5.87863 \times 10^{12} \, \text{miles} \approx 8.34564 \times 10^{12} \, \text{miles}\]

Final Results

After performing all the necessary conversions, here are the results:(a) 32.4 yd = 2964.656 cm(b) \(3.0 \times 10^{10} \text{ cm/s} = 9.8425 \times 10^{8} \text{ ft/s}\)(c) 1.42 light-years = \(8.34564 \times 10^{12} \text{ miles}\)

Key concepts

yard to centimeters conversion

When converting measurements from one unit to another, it's crucial to understand the conversion factors involved. For converting yards to centimeters, you start by converting yards to meters. The conversion factor here is 1 yard equals 0.9144 meters. Once you have the measurement in meters, you can easily convert it to centimeters, as 1 meter equals 100 centimeters.

For example, if you have 32.4 yards, you first multiply by 0.9144 to convert to meters:\[32.4 \, ext{yd} \times 0.9144 \, \text{m/yd} = 29.64656 \, \text{m}\]

Then, convert the meters to centimeters by multiplying by 100:\[29.64656 \, \text{m} \times 100 \, \text{cm/m} = 2964.656 \, \text{cm}\]

This method of conversion is very handy, as it breaks down the process into simple, manageable steps, ensuring accuracy and ease in unit conversions.

centimeter per second to feet per second

Speed conversions are common in physics and chemistry, where precision is key. When converting between centimeters per second and feet per second, you will use the conversion factor that 1 foot equals 30.48 centimeters. This factor helps bridge the gap between the metric and imperial systems of measurement.

For instance, to convert the speed of \(3.0 \times 10^{10} \, \text{cm/s}\) to feet per second, you multiply by the conversion factor: \[3.0 \times 10^{10} \, \text{cm/s} \times \left( \frac{1 \, \text{ft}}{30.48 \, \text{cm}} \right) \approx 9.8425 \times 10^{8} \, \text{ft/s}\]

This straightforward conversion allows you to convert velocities measured in centimeters per second into the more widely used feet per second within the engineering and physics context, keeping measurements consistent with system requirements.

light-year distance calculation

The light-year is a unit of distance used mainly in astronomy, representing the distance light travels in one year. To understand this, you need to consider the speed of light, which is \(3.00 \times 10^{8} \, \text{m/s}\). The total distance light travels in a year can be calculated by multiplying this speed by the total number of seconds in a year. This is essential for understanding vast astronomical distances.

To compute the distance in meters for one year, multiply the speed of light by the seconds in a year:\[3.00 \times 10^{8} \, \text{m/s} \times 365 \times 24 \times 3600 \, \text{s} = 9.4608 \times 10^{15} \, \text{meters}\]

Then, to convert this distance into miles, use the conversion factor 1 mile equals 1609.34 meters:\[9.4608 \times 10^{15} \, \text{m} \times \left( \frac{1 \, \text{mile}}{1609.34 \, \text{m}} \right) \approx 5.87863 \times 10^{12} \, \text{miles}\]

Next, multiply this result by the number of light-years you want to convert, 1.42 in this case:\[1.42 \times 5.87863 \times 10^{12} \, \text{miles} \approx 8.34564 \times 10^{12} \, \text{miles}\]

This calculation gives you a sense of the incredible distances involved in space and the usefulness of the light-year as a unit of measurement.