Question

How long does it take light to travel: (a) \(1.0 \mathrm{ft}\) (report answer in nanoseconds) (b) \(2462 \mathrm{mi}\), the distance between Los Angeles and New York (report answer in milliseconds) (c) \(4.5\) billion \(\mathrm{km}\), the average separation between the sun and Neptune (report answer in hours and minutes)

Short answer

Light takes approximately: (a) 3.34 nanoseconds to travel 1.0 ft, (b) 13.11 milliseconds to travel 2462 mi, and (c) 4 hours and 15 minutes to travel 4.5 billion km.

Step-by-step solution

Convert feet to meters

To calculate the time it takes for light to travel 1.0 ft, convert feet to meters because the speed of light is given in meters per second. Use the conversion factor: 1 ft = 0.3048 m.

Calculate time in seconds for part (a)

Using the speed of light in a vacuum, which is approximately \(3.0 \times 10^{8} \) m/s, divide the distance in meters by the speed of light to get the time in seconds: Time (s) = Distance (m) / Speed of light (m/s).

Convert seconds to nanoseconds for part (a)

Since 1 second = \(1 \times 10^{9}\) nanoseconds, multiply the time in seconds by \(1 \times 10^{9}\) to convert to nanoseconds.

Convert miles to meters for part (b)

To find out how long it takes light to travel 2462 miles, first convert miles to meters using the conversion factor: 1 mile = 1,609.34 meters.

Calculate time in seconds for part (b)

Just like in part (a), use the speed of light to calculate the time in seconds: Time (s) = Distance (m) / Speed of light (m/s).

Convert seconds to milliseconds for part (b)

Since 1 second = 1000 milliseconds, multiply the time in seconds by 1000 to convert to milliseconds.

Convert kilometers to meters for part (c)

For part (c), convert the average separation distance between the sun and Neptune from kilometers to meters by using the conversion factor: 1 km = 1000 m.

Calculate time in seconds for part (c)

Using the speed of light, calculate the time in seconds: Time (s) = Distance (m) / Speed of light (m/s).

Convert seconds to hours and minutes for part (c)

To convert the time from seconds to hours, divide by the number of seconds in an hour (3600). Then, take the remainder and convert it to minutes by dividing by the number of seconds in a minute (60).

Key concepts

Light Travel Time

Understanding the light travel time is crucial for both physics students and enthusiasts alike. Essentially, it's the duration light takes to cover a specific distance. The speed of light in a vacuum is a constant, approximately 299,792,458 meters per second (m/s). This means that light can circumnavigate the Earth over seven times in just one second! In our exercise examples, such as calculating the time light takes to travel 1.0 ft, the process involves a couple of important steps.

First, as light travels so quickly, the resultant time for short distances can be exceptionally small, leading to the necessity of using units like nanoseconds (one billionth of a second) for practical and precise values. This approach was apparent in part (a) of our exercise, where after converting the distance from feet to meters, the light travel time was best expressed in nanoseconds to accurately measure such an infinitesimal span.

Conversion Factors

When dealing with the speed of light calculations, conversion factors become our best friends. These are used to convert one unit of measurement into another. For instance, as seen in the given steps of our exercise, distances have to be converted into meters because the standard unit for the speed of light is meters per second. This isn't always intuitive because we might measure distances in feet, miles, or kilometers in everyday life. Hence, knowing that 1 ft is 0.3048 m, 1 mile is 1,609.34 m, and 1 km is 1,000 m, helps us translate our distance into the proper unit to use with the speed of light.

This process is not only crucial for physicists but also ingrained in the navigational systems that guide our transportation and telecommunications technologies. Mastery of conversion factors ensures accuracy and cohesiveness across various scientific and engineering disciplines.

Velocity of Light

The velocity of light, often referred to as the speed of light, is a fundamental constant in physics denoted by the letter 'c'. The value of 'c' is approximately 299,792,458 m/s or about 3.00 x 10^8 m/s in scientific notation for ease of calculation, as used in our exercise. This incredible speed is the upper limit for how fast any information or matter can travel through space, according to Einstein's theory of relativity.

Recognizing this uniform limit is key to understanding various physical phenomena and is the cornerstone of modern physics, influencing our concepts of space and time.

Distance-Time Relationship

The distance-time relationship is a fundamental concept that describes how far something travels over a period. The formula used to find this is: Time = Distance / Speed. This equation implies that the time taken to cover a distance is directly proportional to the distance and inversely proportional to the speed. Hence, if you increase the distance, travel time increases, and if you increase the speed, travel time decreases.

In the context of light, as its speed is extraordinarily high, the resulting travel time for distances we encounter on Earth and within our solar system is extremely small. Nevertheless, in cosmic terms, even light takes significant time to travel between stars and galaxies, which is why we sometimes see heavenly bodies as they were ages ago – we're looking at the light that left them many years in the past.