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## Homework Statement

A satellite moves around the Earth in a circular orbit of radius r.

(a) What is the speed vi of the satellite?

(b) Suddenly, an explosion breaks the satellite into two pieces, with masses m and 4m. Immediately after the explosion, the smaller piece of mass m is stationary with respect to the Earth and falls directly toward the Earth. What is the speed v of the larger piece immediately after the explosion?

(c) Because of the increase in its speed, this larger piece now moves in a new elliptical orbit. Find its distance away from the center of the Earth when it reaches the the other end of the ellipse

## Homework Equations

ΔE = ΔK + ΔU = 0

ΔƩmv = 0

## The Attempt at a Solution

The apparently correct solution involves using the conservation of momentum. I tried to use the conservation of kinetic energy. My reasoning is that if the smaller piece of mass m is basically at rest after the explosion, it no longer has kinetic energy, thus all of the final kinetic energy is in the moving piece of mass 4m. I assume that the original mass is 5m. Equating the original kinetic energy of the object before the explosion to the kinetic energy of the smaller object, I got the following result:

[tex] K_i=\dfrac{1}{2}m_iv_i^2=\dfrac{5}{2}m\left(\dfrac{GM_E}{R_E+h}\right) [/tex]

[tex] K_f=2mv_f^2 [/tex]

[tex] \dfrac{5}{2}m(\dfrac{GM_E}{R_E+h})=2mv_f^2 [/tex]

[tex] v_f=\sqrt{\dfrac{5GM_E}{4(R_E+h)}} [/tex]

This is apparently wrong, but I'm not sure why. One thing that troubles me is that I do not account for potential gravitational energy. Keep in mind the correct answer, according to the solution's manual, is:

[tex] v_f=\dfrac{5}{4}\sqrt{\dfrac{GM_E}{R_E+h}} [/tex]

Also I do not know how to begin part (c). Intuitively I'm thinking that if the satellite is on the "other side" of an elliptical orbit, it's distance from the surface of the Earth would be the same as it's distance at it's starting point...suggestions?