What is the density in Mg/m^3 of a substance with a density of 0.14 lb/in^3? What is the velocity in m/s of a vehicle traveling at 70 mph?

To convert the density from lb/in^3 to Mg/m^3 (where Mg stands for megagrams which is equivalent to metric tonnes), you need to follow two main steps: First, convert pounds (lb) to megagrams (Mg), and inches cubed (in^3) to meters cubed (m^3).

1. Convert lb to Mg: 1 pound is approximately 0.00045359237 megagrams. So, 0.14 lb is 0.14 * 0.00045359237 = 0.0000635027 Mg.

2. Convert in^3 to m^3: There are 61,023.7 cubic inches in a cubic meter (since a meter is 39.3701 inches, and cubing this conversion factor gives us (39.3701 inches/m)^3 or approximately 61,023.7 in^3/m^3). So, for density in Mg/m^3, we divide the mass in Mg by the volume in m^3. Since we have the mass in Mg for one cubic inch, we scale that mass to one cubic meter by multiplying by the number of cubic inches in a cubic meter. Density in Mg/m^3 = 0.0000635027 Mg/in^3 * 61,023.7 in^3/m^3 = 3.8754 Mg/m^3 (rounded to four decimal places).

For the velocity conversion:

1. Convert mph to m/s: First, understand that 1 mile is equal to 1,609.34 meters, and there are 3,600 seconds in an hour (since there are 60 seconds per minute and 60 minutes per hour). So, to convert 70 mph to m/s, you perform the following calculation: 70 miles/1 hour * 1,609.34 meters/1 mile * 1 hour/3,600 seconds = 111,853.8 meters/3,600 seconds ≈ 31.07 m/s (rounded to two decimal places).

Hence, the density of the substance is approximately 3.8754 Mg/m^3, and the velocity of the vehicle is approximately 31.07 m/s.

Extra: When we work with units, it's crucial to keep track of the conversion factors for each unit. In the metric system, the base units for mass and length are kilograms (kg) and meters (m), respectively. However, in the engineering and scientific fields, we often utilize multiples of these units like megagrams (Mg) and cubic meters (m^3).

In everyday life, especially in the United States, people might use pounds (lb) for mass and inches (in) for length. In many scientific and engineering applications, though, it's common to use metric units because they're more standardized and widely accepted internationally.

Additionally, velocity is a vector quantity, meaning it has both magnitude and direction. Here we're only considering the magnitude when we do the conversion, and the direction of the velocity would depend on the specific situation. The choice of units for velocity depends on the context; in the United States, miles per hour (mph) is commonly used, while the metric system uses meters per second (m/s). The conversion factors allow you to switch between units, adapting your calculation to the context required.

In both of these conversions, understanding how to multiply the given value by the appropriate conversion factors to cancel out the original units and introduce the desired units is the key to successful unit conversion.