Unit conversion [25 points] Conversion formula: 1 km = 1000 m 1 m = 100 cm 1 mi = 1.61 km 1 hr = 3600 s 1 kg = 1000 g a) [8 points] The speed of light in vacuum is 3.00 x 10
m/s. What is this speed in kilometers per hour? b) [7 points] What is the speed of light in vacuum in miles per hour? c) [10 points] The density of aluminum is 2.70 g/cm
. What is this density in kilograms per cubic meter?
The Correct Answer and Explanation is:
Problem 1: Unit Conversion [25 points]
a) Speed of light in km/hr
The speed of light in vacuum is:3.00×108 m/s3.00 \times 10^8 \ \text{m/s}3.00×108 m/s
Step 1: Convert meters to kilometers1 km=1000 m⇒11000 km=1 m1 \ \text{km} = 1000 \ \text{m} \Rightarrow \frac{1}{1000} \ \text{km} = 1 \ \text{m}1 km=1000 m⇒10001 km=1 m3.00×108 m/s=3.00×1081000=3.00×105 km/s3.00 \times 10^8 \ \text{m/s} = \frac{3.00 \times 10^8}{1000} = 3.00 \times 10^5 \ \text{km/s}3.00×108 m/s=10003.00×108=3.00×105 km/s
Step 2: Convert seconds to hours1 hr=3600 s1 \ \text{hr} = 3600 \ \text{s}1 hr=3600 s3.00×105 km/s×3600 s/hr=1.08×109 km/hr3.00 \times 10^5 \ \text{km/s} \times 3600 \ \text{s/hr} = 1.08 \times 10^9 \ \text{km/hr}3.00×105 km/s×3600 s/hr=1.08×109 km/hr
Answer (a):1.08×109 km/hr\boxed{1.08 \times 10^9 \ \text{km/hr}}1.08×109 km/hr
b) Speed of light in mi/hr
1 mi=1.61 km1 \ \text{mi} = 1.61 \ \text{km}1 mi=1.61 km1.08×109 km/hr÷1.61=6.71×108 mi/hr1.08 \times 10^9 \ \text{km/hr} \div 1.61 = 6.71 \times 10^8 \ \text{mi/hr}1.08×109 km/hr÷1.61=6.71×108 mi/hr
Answer (b):6.71×108 mi/hr\boxed{6.71 \times 10^8 \ \text{mi/hr}}6.71×108 mi/hr
c) Density of aluminum in kg/m³
Given:2.70 g/cm32.70 \ \text{g/cm}^32.70 g/cm3
Step 1: Convert grams to kilograms1 kg=1000 g⇒2.70 g=2.701000=0.0027 kg1 \ \text{kg} = 1000 \ \text{g} \Rightarrow 2.70 \ \text{g} = \frac{2.70}{1000} = 0.0027 \ \text{kg}1 kg=1000 g⇒2.70 g=10002.70=0.0027 kg
Step 2: Convert cubic centimeters to cubic meters1 m=100 cm⇒(100 cm)3=106 cm31 \ \text{m} = 100 \ \text{cm} \Rightarrow (100 \ \text{cm})^3 = 10^6 \ \text{cm}^31 m=100 cm⇒(100 cm)3=106 cm31 cm3=10−6 m31 \ \text{cm}^3 = 10^{-6} \ \text{m}^31 cm3=10−6 m3
So:2.70 g/cm3=0.0027 kg10−6 m3=2700 kg/m32.70 \ \text{g/cm}^3 = \frac{0.0027 \ \text{kg}}{10^{-6} \ \text{m}^3} = 2700 \ \text{kg/m}^32.70 g/cm3=10−6 m30.0027 kg=2700 kg/m3
Answer (c):2700 kg/m3\boxed{2700 \ \text{kg/m}^3}2700 kg/m3
Explanation (300+ words)
Unit conversion is a fundamental skill in physics and engineering that allows measurements to be understood and compared across different systems. The process involves multiplying the original measurement by conversion factors that equal 1, ensuring that units cancel appropriately and yield the desired units.
Part (a) involves converting the speed of light from meters per second (m/s) to kilometers per hour (km/hr). Since 1 kilometer equals 1000 meters, dividing by 1000 changes meters to kilometers. Then, multiplying by 3600 converts seconds to hours. This two-step process transforms the units from m/s to km/hr accurately. The result shows how extraordinarily fast light travels—over a billion kilometers per hour.
Part (b) requires converting the result from part (a) into miles per hour. Using the conversion factor 1 mile = 1.61 kilometers, the value in kilometers per hour is divided by 1.61 to yield miles per hour. This gives a better sense of scale for those familiar with imperial units and demonstrates the incredible speed of light in a different unit system.
Part (c) focuses on converting a density measurement from grams per cubic centimeter (g/cm³) to kilograms per cubic meter (kg/m³), a common SI unit. To convert grams to kilograms, divide by 1000. To convert cubic centimeters to cubic meters, recognize that 1 m=100 cm1 \text{ m} = 100 \text{ cm}1 m=100 cm, so 1 m3=106 cm31 \text{ m}^3 = 10^6 \text{ cm}^31 m3=106 cm3. Therefore, each cm³ is 10−610^{-6}10−6 m³, and the final conversion is accomplished by dividing kilograms by 10−610^{-6}10−6, effectively multiplying the density by 10610^6106. This results in a commonly used density of 2700 kg/m³ for aluminum.
