A 1024-kg blue ball is dropped from an initial z-position of 3.9 x 106 m through the center of a planet with radius 6.9 x 106 m. If the mass of the planet is 42 x 1015 kg, measure the displacement of the ball at time t = 8 s?

A meteoroid is moving towards a planet. It has mass m = 0.54×109 kg and speed v1 = 4.7×107 m/s at distance R1 = 1.6×107 m from the center of the planet. The radius of the planet is R = 0.78×107 m. The mass of the planet is M = 5.6×1025kg. There is no air around the planet.  a)Enter an expression for the total energy E of the meteoroid at R, the surface of the planet, in terms of defined quantities and v, the meteoroid’s speed when it reaches the planet’s surface. Select from the variables below to write your expression. Note that all variables may not be required.α, β, θ, d, g, G, h, m, M, P, R, R1, t, v, v1 b)Enter an expression for v, the meteoroid’s speed at the planet’s surface, in terms of G, M, v1, R1, and R. c)Calculate the value of v in meters per second.

Aliens declare war on Earth by dropping an asteroid on our planet.  The asteroid starts at rest from a distance of 37 x 106 m from the center of Earth.  Calculate the speed with which the asteroid hits Earth's surface, in km/s.  Use that G = 6.7 x 10-11 N m2 / kg2, MEarth = 6 x 1024 kg, and REarth = 6 x 106 m. (Please answer to the fourth decimal place - i.e 14.3225)

The free-fall acceleration on the surface of Jupiter is about two and one half times that on the surface of the Earth. The radius of Jupiter is about 11.0 RE (RE = Earth's radius = 6.4  106 m). Find the ratio of their average densities, ?Jupiter/?Earth.