Question

For the following exercises, use Kepler's Law, which states that the square of the time, T (measured in Earth years), required for a planet to orbit the Sun varies directly with the cube of the mean distance, a (measured in billions of miles), that the planet is from the Sun. a. Use Mars's time of 2 and mean distance of 0.142 billion miles, find the equation relating T and a. (Enter an equation in terms of T and a. Round numbers to the nearest tenth.) b. Use the result from part a. to determine the time required for Pluto to orbit the Sun if its mean distance is 3.67 billion miles. It takes Pluto approximately years to orbit the Sun. (Round your answer to 3 decimal places.)

Answer 1

Answer:

Data provided in the question:

According to Kepler's Law, the square of the time "T" (measured in Earth years) required for a planet to orbit the sun varies directly with the cube of the mean distance "a" (measure in billions of miles) that the planet is from the sun, i.e.

T? = kg

where "k" is the constant.

a.

Time required by planet Mars to orbit the sun, T_m = 2

Mean distance of planet Mars from the sun, a_m = 0.142 billion miles.

On substituting the above values in the Kepler's Law equation, we get -

(2)2 = k(0.142)

4= k(0.002863)

= 139713

Therefore, we can write the equation for Kepler's Law as -

T2 = (1397.1)

b.

Mean distance of planet Pluto from the sun, a_p = 3.67 billion miles.

Time required by planet Pluto to orbit the sun, T_p = ?

We can use the equation of the Kepler's Law for calculating the time required by planet Pluto to orbit the sun as -

Tº = (1397.1) (3.67)

T2 = (1397.1) (49.430863)

T2 = 69059.8587

::T=V69059.8587

  

= 262,792

Thus, the time required by planet Pluto to orbit the sun is 262.792 Earth years.