If Earth has an orbital period of 365 days around the Sun, what would be the orbital period of a BRAND NEW planet that was exactly twice the orbital radius as Earth Use the equation r3/t2 of object 1= r3/t2 of object 2 Start with showing your Equation with known values in place. Show your steps
The equation is r1^3/t1^2 = r2^3/t2^2, where r1 and r2 are the orbital radii of object 1 and object 2 respectively, and t1 and t2 are the orbital periods of object 1 and object 2 respectively.
Given:
r1 = orbital radius of Earth = 1
t1 = orbital period of Earth = 365 days
r2 = 2 * r1 = 2 * 1 = 2
Substituting the known values into the equation, we have:
(1)^3/(365)^2 = (2)^3/t2^2
Simplifying the equation:
1/365^2 = 8/t2^2
Cross-multiplying:
t2^2 = (365^2) * 8
Taking the square root of both sides:
t2 = √((365^2) * 8)
Calculating this value:
t2 ≈ √(133225 * 8)
t2 ≈ √1065800
t2 ≈ 1032.136
Therefore, the orbital period of the brand new planet would be approximately 1032.136 days.
Given:
r1 = orbital radius of Earth
t1 = orbital period of Earth
r2 = 2 * r1 (orbital radius of the new planet)
To find:
t2 = orbital period of the new planet
The equation relating the orbital radius and period of two objects is:
r1^3 / t1^2 = r2^3 / t2^2
Substituting the known values, we have:
(r1^3 / t1^2) = (r2^3 / t2^2)
Replacing r2 with 2 * r1:
(r1^3 / t1^2) = ((2 * r1)^3 / t2^2)
Expanding (2 * r1)^3:
(r1^3 / t1^2) = (8 * r1^3 / t2^2)
Cross-multiplying to solve for t2^2:
r1^3 * t2^2 = 8 * r1^3 * t1^2
Dividing both sides by r1^3:
t2^2 = 8 * t1^2
Taking the square root of both sides to solve for t2:
t2 = √(8 * t1^2)
Therefore, the orbital period of the new planet, t2, would be √(8 * t1^2).