Problem 77 - Algebra 2

Question

Earth is an average of \(1.496 \times 10^{8}\) kilometers from the Sun. If light travels \(3 \times 10^{5}\) kilometers per second, how long does it take sunlight to reach Earth?

Step-by-Step Solution

Answer

Sunlight takes approximately 499 seconds to reach Earth.

Step 1: Understand the Problem

We need to find out how long it takes for sunlight to travel from the Sun to Earth. This involves calculating the time it takes for light to cover the distance between the Earth and the Sun, given the speed of light.

Step 2: Apply the Distance Formula

The formula to calculate time is: \( \text{time} = \frac{\text{distance}}{\text{speed}} \). We have the distance from Earth to the Sun, which is \(1.496 \times 10^{8}\) kilometers, and the speed of light, which is \(3 \times 10^{5}\) kilometers per second.

Step 3: Insert Values into the Formula

Substitute the known values into the time formula: \[ \text{time} = \frac{1.496 \times 10^{8}}{3 \times 10^{5}} \].

Step 4: Perform the Calculation

Divide \(1.496 \times 10^{8}\) by \(3 \times 10^{5}\). To do this, divide the coefficients and subtract the exponents of 10.\[ \frac{1.496}{3} \approx 0.4987 \] and \(10^{8-5} = 10^{3}\). Thus, \(0.4987 \times 10^{3} = 498.7\).

Step 5: Interpret the Result

Since time is usually expressed in whole seconds for practical purposes, round \(498.7\) to \(499\) seconds. Therefore, it takes approximately 499 seconds for sunlight to reach Earth.

Key Concepts

Distance Formula

The distance formula is a crucial tool in many areas of science and everyday life. When trying to determine how long it takes for something to travel from point A to point B, knowing the speed at which it is moving and the distance between the two points allows us to calculate the time it will take.

The formula can be expressed as:

• \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)

This formula is simple yet powerful. It tells us that the time it takes to travel a certain distance is directly proportional to the distance itself and inversely proportional to the speed of travel.

For our solar system example, we know the distance between the Earth and the Sun and the speed of light. By plugging these values into the formula, we can find out how long light takes to travel from the Sun to Earth.Let’s consider this with our example:

• Distance (Sun to Earth): \(1.496 \times 10^8\) kilometers
• Speed of Light: \(3 \times 10^5\) kilometers per second

Using the distance formula helps us solve the problem with ease.

Time Calculation

Calculating time is about finding out how long it takes for an event or movement to occur. In our exercise, we are curious about how long it takes for light to travel from the Sun to Earth.

The time calculation using the formula

• \( \text{time} = \frac{1.496 \times 10^8}{3 \times 10^5} \)

involves dividing the distance (\(1.496 \times 10^8\) km) by the speed of light (\(3 \times 10^5\) km/s). First, we divide the numerical coefficients, which gives:

• \( \frac{1.496}{3} \approx 0.4987 \)

Then, subtract the exponents of 10:

• \(10^{8-5} = 10^3\)

Combining these results, we get:

• \(0.4987 \times 10^3 = 498.7\) seconds

Rounding this off, we find it takes approximately 499 seconds, which is about 8 minutes and 19 seconds, for light to reach Earth. This is a practical calculation often used in astronomy to understand light travel time.

Astronomical Distances

The universe is vast, and astronomical distances can be incredibly large. When we talk about distances such as those between the Earth and the Sun, typical units like meters or kilometers are used, but even these can result in enormous numbers.

Astronomical distances require consideration of both scale and precision. The average distance from the Earth to the Sun is a staggering \(1.496 \times 10^8\) kilometers, which necessitates using scientific notation to handle such large numbers conveniently.

This distance is referred to as an "Astronomical Unit" (AU), which is a standard measure used in astronomy to describe distances within our solar system:

• 1 AU is approximately equal to the average Earth-Sun distance.
• Using AU simplifies the comparison and communication of distances within our solar system.

Understanding astronomical distances helps in the study of celestial events and determining the travel time of light, critical in observing and understanding the universe.