What is the moon slowly receding from us and by how much each year?

The moon is receding from the Earth by approximately 3.8 cm per year. Earth's mass is 5.98 x 10^24 kg, and its?

• radius is 6.38 x 10^6 m. The Moon's mass is 7.3 x 10^22 kg, its radius is 1.79 x 10^6 m, and its orbital period around Earth is 27.3 days. The current average distance between the 2 surfaces is 3.85 x 10^8 m. Assume that neither body gains or loses mass and that the recession continues at a rate of 3.8 cm per year. a.) Approximately how much will the gravitational attraction between the Moon and Earth change between now and 499 million years from now? b.) Approximately how long, in present Earth-days, will it take the Moon to orbit Earth 499 million years from now?

• Answer:

  a) The increase in distance is 499*10^6 * 0.038 m = 1.90*10^7 m As a % of the present distance this is 100* 0.19/3.85 = 4.935% Gravitational force is proportional to 1/ distance^2, so the % reduction in gravitational attraction will be: 100 * (1 - 1/(1.04935)^2) = approx 9.2% Use Newton's Law of gravitation f = G .Me . Mm / r^2 if you need the actual forces. b) According to Kepler's 3rd Law (orbit period)^2 is proportional to radius^3 So the period will increase by 100 *(1 - sqrt(1.04935)^3) = approx 7.5% In days: 27.3 days * 1.075 = 29.3 days