Planetary orbits. The formula T(x)=x^{\frac{3}{2}} calculates the time in
tabita57i
Answered question
2021-09-22
Planetary orbits. The formula calculates the time in years that it takes a planet to orbit the sun if the planet is x times farther from the sun than earth is.
a) Find the inverse of T
b) What does the inverse of T calculate?
Given:
The time in years that it takes a planet to orbit the sun if the planet is x times farther from the sun than earth is given by
Answer & Explanation
Tuthornt
Skilled2021-09-23Added 107 answers
Formula used:
The formula which converts planet orbital radius to orbital period. If and f(x) is a bijective mapping (both one-to-one and surjective) then its inverse exists and thus .
Calculation:
a) Consider the given formula with , which is an increasing function satisfying bijectivity. Now, raising both the sides to power we get . This gives,
b) From part (a) we have seen . Thus inverse of T calculates orbit radius as a function of orbital period.
Conclusion:
Thus, we found out that is the inverse of T and it calculates orbit radius as a function of orbital period. We can also explain it in another way. If planet orbital period function is represented by , its inverse can be expressed as . For T(x), the argument x represents the number of times the planet with orbital period T is farther from sun than Earth is.