Question

Earth II is a planet in a distant solar system which is earth-like, but a little smaller than our earth. In the distant future, Earth II has been settled by bold travelers from our home planet. Very far from Earth II (effectively at R=[infinity]), a spacecraft has run out of fuel and its kinetic energy is zero. If only the gravitational force of Earth II were to act on the spacecraft (i.e., neglect the forces from the sun and other solar system objects), the spacecraft would eventually crash into Earth II. The mass of Earth II is Me=3.890×10²⁴ kg and its radius is Re=5.530×10⁶ m . Neglect air resistance throughout this problem, since the spacecraft is primarily moving through the near vacuum of space.

Find the speed (se) of the spacecraft when it eventually crashes into Earth II.

Answer 1

The crash speed of the spacecraft onto Earth II is determined by using the principles of conservation of energy, converting gravitational potential energy into kinetic energy.

Step-by-step explanation:

The speed (se) of the spacecraft when it crashes into Earth II is found using conservation of energy, specifically gravitational potential energy and kinetic energy. When the spacecraft is very far from Earth II, it has a certain amount of gravitational potential energy which gets converted into kinetic energy as it falls towards Earth II.At the starting point, the gravitational potential energy is U = -G * Me * m / R, where G is the gravitational constant, Me is the mass of Earth II, m is the mass of the spacecraft, and R is the distance from the center of Earth II which is effectively infinite at the beginning. Kinetic energy is given by K = (1/2) * m * se^2.

As the spacecraft falls, energy conservation implies that the sum of potential and kinetic energy remains the same. When the spacecraft hits the surface of Earth II, all of the potential energy has been converted into kinetic energy.To find the velocity at impact, we can set up the equation G * Me * m / Re = (1/2) * m * se^2 and solve for se, resulting in se = √(2 * G * Me / Re). Plugging in the values G = 6.674 × 10^-11 N·m^2/kg^2, Me = 3.890 × 10^24 kg, and Re = 5.530 × 10^6 m, we get the impact speed se.