Very far from earth (at R- oo), 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 (ie., 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 Find the speed s of the spacecraft when it crashes into the earth Express the speed in terms of M, Re, and the universal gravitational constant G.

Speed of the spacecraft right before the collision: .

Assumption: the earth is exactly spherical with a uniform density.

Explanation:

This question could be solved using the conservation of energy.

The mechanical energy of this spacecraft is the sum of:

Let denote the mass of this spacecraft. At a distance of from the center of the earth (with mass ), the gravitational potential energy () of this spacecraft would be:

.

Initially, (the denominator of this fraction) is infinitely large. Therefore, the initial value of will be infinitely close to zero.

On the other hand, the question states that the initial kinetic energy () of this spacecraft is also zero. Therefore, the initial mechanical energy of this spacecraft would be zero.

Right before the collision, the spacecraft would be very close to the surface of the earth. The distance between the spacecraft and the center of the earth would be approximately equal to , the radius of the earth.

The of the spacecraft at that moment would be:

.

Subtract this value from zero to find the loss in the of this spacecraft:

Assume that gravitational pull is the only force on the spacecraft. The size of the loss in the of this spacecraft would be equal to the size of the gain in its .

Therefore, right before collision, the of this spacecraft would be:

.

On the other hand, let denote the speed of this spacecraft. The following equation that relates and to :

.

Rearrange this equation to find an equation for :

.

It is already found that right before the collision, . Make use of this equation to find at that moment:

.

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waves moving from a fast medium to a slow medium will experience a reflection as light travels more slowly than red light. total internal reflection occurs when the critical angle is.