A space probe is studying a newly discovered spherical planet of radius R . The probe measures the gravitational acceleration at a height of R 2 above the planet's surface to be a₁ . Later, the probe moves to a higher orbit at a height of 2R above the surface, where the gravitational acceleration is measured to be a₂ . What is the ratio a₁ a₂ ?

Step-by-step solution

The gravitational acceleration at a distance r from the center of a planet is given by g = GM r^2 . For the first orbit, the height is h₁ = R 2 , so the distance from the center is r₁ = R + R 2 = 3R 2 . The acceleration is a₁ = GM ( 3R 2 )^2 = 4GM 9R^2 . For the second orbit, the height is h₂ = 2R , so the distance from the center is r₂ = R + 2R = 3R . The acceleration is a₂ = GM (3R)^2 = GM 9R^2 . The ratio of the accelerations is a₁ a₂ = 4GM 9R^2 GM 9R^2 = 4 . Thus, the ratio is 4 : 1 . Answer: 4 : 1