Question

Earth is approximately 93 000 000 mi from the sun. It revolves around the sun, in an almost circular orbit, in about 365 days. Calculate the linear speed, in miles per hour, of Earth in its orbit. Give your answer to the nearest hundredth.

Short answer

Earth's linear speed is approximately 66,627.36 mph.

Step-by-step solution

Calculate the circumference of Earth's orbit

Use the formula for the circumference of a circle, which is \[ C = 2\pi r \], where \( r \) is the radius of the orbit. Here, \( r \) is 93,000,000 miles. Compute the circumference as follows:\[ C = 2 \pi \times 93{,}000{,}000 \approx 584{,}336{,}233.57 \text{ miles} \].

Calculate the total time in hours for one year

Determine the total number of hours in one year. Given that there are 365 days in a year and 24 hours in a day, the total time is\[ 365 \times 24 = 8760 \text{ hours} \].

Calculate the linear speed

Use the formula for speed, which is \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \), to find the linear speed. The distance covered by Earth in one year is the circumference of its orbit, and the time is 8760 hours:\[ \text{Speed} = \frac{584{,}336{,}233.57}{8760} \approx 66{,}627.36 \text{ miles per hour} \].

Key concepts

Circumference of a Circle

To understand how we calculate the circumference of Earth's orbit, we need to know the formula for the circumference of a circle. The formula is: \( C = 2\text{π}r \), where \( C \) represents the circumference and \( r \) is the radius. Earth's orbit around the sun is almost circular, with a radius \( r \) of approximately 93,000,000 miles. Plugging in the values, we get: \[ C = 2\text{π} \times 93{,}000{,}000 \approx 584{,}336{,}233.57 \text{ miles} \]. This calculation shows us the total distance Earth travels in one complete orbit around the sun. Understanding how to compute the circumference is crucial because it serves as our distance in the speed calculation.

Linear Speed Calculation

Calculating Earth's linear speed in its orbit involves determining how fast Earth travels in a given amount of time. Linear speed can be found using the formula: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \). From the first step, we know the distance (the circumference of Earth's orbit), which is approximately 584,336,233.57 miles. Next, we need to find out the total time. Earth takes about 365 days to complete one orbit, and each day has 24 hours. Therefore, the total time in hours is \365 \times 24 = 8760 \text{ hours}\. Now, plugging in the values into the speed formula, we get: \[ \text{Speed} = \frac{584{,}336{,}233.57}{8760} \approx 66{,}627.36 \text{ miles per hour} \]. This means Earth travels around the sun at a speed of about 66,627.36 miles per hour.

Earth's Orbit Distance

Earth's orbit distance is essentially the circumference of its near-circular path around the sun. Knowing the distance helps us understand planetary motion and the vast scales involved. Earth's distance from the sun varies slightly, but it averages 93,000,000 miles. This average distance is what we use to compute the circumference, and subsequently, the linear speed. By understanding the orbit distance, students can better grasp concepts related to space travel, satellite placement, and gravitational effects. The enormous distance that Earth covers in one year underlines the immense, yet consistent nature of celestial mechanics and highlights the significance of calculating such distances accurately.