To compare the time it takes for light and sound to travel around the world, we need to know the speeds of light and sound and the distance they need to travel.
### Step 1: Determine the speed of light and sound
- The speed of light in a vacuum is approximately [tex]\( 3 \times 10^8 \)[/tex] meters per second (or 300,000 km/s).
- The speed of sound in air at room temperature (20°C or 68°F) is approximately 343 meters per second (or 0.343 km/s).
### Step 2: Convert the distance around the equator to meters
The distance around the equator is given as 40,000 km. To use standard units consistently, we convert this to meters:
[tex]\[ 40,000 \, \text{km} \times 1,000 \, \text{m/km} = 40,000,000 \, \text{m} \][/tex]
### Step 3: Calculate the time it takes for light to travel around the world
To find the time, we use the formula:
[tex]\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \][/tex]
For light:
[tex]\[ \text{Time}_{\text{light}} = \frac{40,000,000 \, \text{m}}{3 \times 10^8 \, \text{m/s}} \][/tex]
[tex]\[ \text{Time}_{\text{light}} = \frac{40,000,000}{300,000,000} \][/tex]
[tex]\[ \text{Time}_{\text{light}} = \frac{4}{30} \][/tex]
[tex]\[ \text{Time}_{\text{light}} = \frac{2}{15} \][/tex]
[tex]\[ \text{Time}_{\text{light}} \approx 0.0001333 \, \text{seconds} \][/tex]
### Step 4: Calculate the time it takes for sound to travel around the world
For sound:
[tex]\[ \text{Time}_{\text{sound}} = \frac{40,000,000 \, \text{m}}{343 \, \text{m/s}} \][/tex]
[tex]\[ \text{Time}_{\text{sound}} \approx 116,617 \, \text{seconds} \][/tex]
### Step 5: Compare the two times
The time for light to travel around the world is approximately 0.0001333 seconds. The time for sound to travel the same distance is approximately 116,617 seconds.
To put that into perspective:
- Light travels around the world almost instantaneously, in just a fraction of a second.
- Sound takes a considerably longer time to travel the same distance. If we convert 116,617 seconds to hours, we get:
[tex]\[ \text{Time}_{\text{sound in hours}} = \frac{116,617}{3600} \][/tex]
[tex]\[ \text{Time}_{\text{sound in hours}} \approx 32.4 \, \text{hours} \][/tex]
### Conclusion
Light takes approximately 0.0001333 seconds to travel around the Earth, while sound takes approximately 32.4 hours. This stark contrast highlights the incredible speed of light compared to sound. The speed of light is essential for our fast communication systems, such as satellite and fiber-optic communications, while the relatively slow speed of sound limits its use over long distances.
