The earth moves around the sun in a nearly circular orbit of radius 1.50 1011 m. During the three summer months (an elapsed time of 7.89 106 s), the earth moves one-fourth of the distance around the sun.
To determine the distance that the Earth moves during the three summer months, we need to find one-fourth of the circumference of the Earth's orbit.
The circumference of a circle is given by the formula:
C = 2πr
Where C is the circumference and r is the radius of the circle.
Given that the radius of the Earth's orbit is 1.50 * 10^11 m, we can plug it into the equation to find the circumference:
C = 2π(1.50 * 10^11)
C = 3.0 * 10^11π meters
To find one-fourth of the circumference, we divide the total circumference by 4:
Distance = (3.0 * 10^11π) / 4
Now, we can calculate the distance that the Earth moves during the three summer months. Let's assume that there are 30 days in a summer month, then the total time in seconds would be:
Total time = 3 months * 30 days/month * 24 hours/day * 60 minutes/hour * 60 seconds/minute
Total time = 7.89 * 10^6 seconds
To calculate the distance traveled by Earth during this time, we can use the formula:
Distance = Speed * Time
However, since the Earth moves in a nearly circular orbit, its speed is not constant. Instead, it is constantly changing according to the principles of orbital motion. To simplify the calculation, we will assume a constant speed derived from the average angular speed.
The average angular speed can be determined by dividing the fraction of the Earth's orbit covered by the time period:
Average angular speed = (1/4) * (2π) / (7.89 * 10^6)
Now we can calculate the distance traveled by Earth during the three summer months:
Distance = Average angular speed * radius
Distance = (Average angular speed) * (1.50 * 10^11 m)
By substituting the value of the average angular speed into the equation, we can calculate the distance traveled by Earth during the three summer months.