Recently, Asteroid 2012 DA14 came within 34,200 km from the center of the earth at its point of closest approach. If the moon goes around the earth once every 27.5 days, what is the ratio of the distance of closest approach of DA14 to the radius of the orbit of the moon?

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To find the ratio of the distance of closest approach of Asteroid 2012 DA14 to the radius of the orbit of the moon, we need to determine the respective distances.

First, let's find the radius of the moon's orbit. Since the moon goes around the Earth once every 27.5 days, we can use this information to calculate the moon's orbital period.

The orbital period of the moon (T) is 27.5 days, which can be converted to hours by multiplying it by 24 since there are 24 hours in a day:

T = 27.5 days * 24 hours/day = 660 hours

Next, we'll use Kepler's Third Law of Planetary Motion, which states that the ratio of the average distance of an object from its center to the cube root of its orbital period is constant. This can be written as:

r^3 / T^2 = k

Where r is the distance from the center of the Earth to the moon's orbit, T is the orbital period of the moon, and k is the constant ratio.

Rearranging the equation to solve for r gives us:

r = (T^2 * k)^(1/3)

Now, we need to convert the orbital period into seconds to be consistent with the units used for the distance of closest approach. Since there are 60 seconds in a minute and 60 minutes in an hour:

T_seconds = 27.5 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute = 2,376,000 seconds

Plugging in the values, the radius of the moon's orbit (r) is:

r = (2,376,000 seconds^2 * k)^(1/3)

Next, we can find the distance of closest approach of Asteroid 2012 DA14, which is given as 34,200 km.

To compare the two distances, we need them to be in the same unit. Since the radius of the moon's orbit is calculated in meters, let's convert 34,200 km to meters.

34,200 km = 34,200,000 meters

Now, we can calculate the ratio:

Ratio = (distance of closest approach of DA14) / (radius of the moon's orbit)

Ratio = 34,200,000 m / r

To determine the final value of the ratio, we need to know the value of the constant k, which depends on the mass of the Earth and the gravitational constant. Since the mass of the Earth is not provided, we can't calculate a specific value for the ratio without this information.

However, using this method, you can substitute in the appropriate constant value for k and obtain the ratio once you have the mass of the Earth.