Question

Very far from earth (at R=[infinity]), a spacecraft has run out of fuel and its kinetic energy is zero. If only the gravitational force of the earth were to act on it (i.e., neglect the forces from the sun and other solar system objects), the spacecraft would eventually crash into the earth. The mass of the earth is Me and its radius is Re. Neglect air resistance throughout this problem, since the spacecraft is primarily moving through the near vacuum of space.

Now find the spacecraft's speed when its distance from the center of the earth is R=αRe, where the coefficient α≥1. Express the speed in terms of se and α.

Answer 1

To find the spacecraft's speed when its distance from the center of the earth is R=αRe, we can use the conservation of energy equation and the equation for escape velocity. The orbital speed can also be calculated using the equation for balancing gravitational force and centripetal force.

Step-by-step explanation:

To find the spacecraft's speed when its distance from the center of the earth is R=αRe, we can use the conservation of energy equation. At the initial position, the object is at a distance of the Earth's radius of orbit and has an initial speed of 30 km/s. At the final position, the object has come to rest and its distance from the center of the Earth is αRe. By setting the total energy equal to zero and solving for the final velocity, we can find the spacecraft's speed.

The escape velocity from the surface of an astronomical body can be obtained by setting the total energy equal to zero. At the surface of the body, the object is located at r₁ = R and it has escape velocity v₁ = Uesc. It reaches r2 = ∞ with velocity v2 = 0. Substituting into the equation, we can solve for the escape velocity in terms of G, M, and R.

The orbital speed of a satellite can be calculated by solving for the speed that balances the gravitational force and the centripetal force. The mass of the satellite cancels out, simplifying the equation. We can use this equation to find the orbital speed of the spacecraft when its distance from the center of the Earth is R=αRe.