A package of mass 10 kg sits at the equator of an airless asteroid of mass 9.0 1

Question

A package of mass 10 kg sits at the equator of an airless asteroid of mass 9.0 1020 kg and radius 3.5 105 m. We want to launch the package in such a way that it will never come back, and when it is very far from the asteroid it will be traveling with speed 211 m/s. We have a large and powerful spring whose stiffness is 3.0 105 N/m. How much must we compress the spring?

Explanation / Answer

he total energy of package is given by: KE + PE = constant We have one "point": (@ the equator) KE = 1/2mv² (v = 37m/s) PE = PEg + PEs (PEg = potential energy from gravity, PEs = potential energy from spring) PEg = GmM/r (m = mass of object, M = mass of asteroid, r is the radius, G - gravitational constant) PEs = 1/2kx² (k = spring constant, x = unknown compression) Now here's the second point: (@ far away from asteroid) KE = 1/2mv² (v = 240m/s) PE = 0 (both gravitational AND spring potential are non-existent) KEf = KE0 + PEg + PEs It's probably easier to think of (so that we can take out the sign issues): ?E = ?KE + ?PE = 0 (by conservation of energy) ?KE is straightforward (just calculate the two kinetic energies and subtract) ?PE = ?PEg + ?PEs So should either or both of these be negative or positive???? Simple, going OUT in space should "subtract" from kinetic energy (i.e. what goes up must come down, so the velocity slows down until it starts coming back down), the ?PEg should be positive (because that would require a drop in KE to "cancel" out the increase. ?PEs on the other hand should ADD to the final kinetic energy, so this should be negative (so that KE has to INCREASE to offset this decrease)!! Explanation: PEg = -GmM/r PEs = 1/2kx² --> so if the final for BOTH is 0, then we have: ?PEg = 0 - -GmM/r = GmM/r ?PEs = 0 - (1/2kx²) = -1/2kx² So you have the following equation: ?KE + GmM/r - 1/2kx² = 0 --> 1/2kx² = GmM/r + ?KE Everything on the right, you can calculate with the given values, so just multiply by 2, divide by k, then take the square root to find x. Notice that the right-hand-side MUST be positive, since ?KE is definitely positive.

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