Question

In: Physics

Zero, a hypothetical planet, has a mass of 5.0 x 1023 kg, a radius of 3.1...

Zero, a hypothetical planet, has a mass of 5.0 x 1023 kg, a radius of 3.1 x 106 m, and no atmosphere. A 10 kg space probe is to be launched vertically from its surface. (a) If the probe is launched with an initial kinetic energy of 5.0 x 107 J, what will be its kinetic energy when it is 4.0 x 106 m from the center of Zero? (b) If the probe is to achieve a maximum distance of 8.0 x 106 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?

Solutions

Expert Solution

We can use the conservation of mechanical energy. Initial total energy is equal to final total energy.
At the surface o f the planet, the initial kinetic energy of the probe was Ke and the potential energy is U
Ei = KE + U = 5.0 x 107 J + G M m / R
Where M is the mass of the planet, m is the mass of the probe and R is the radius of the radius of the planet
The final mechanical energy of the planet is
Ef = KE - G Mm / r
Where r is the distance of the planet from the probe.
r = 4.0 x 106 m
Ei = Ef
5.0 x 107 J - G M m / R = KE - G Mm / r
KE = 5.0 x 107 J - G M m (1/ R - 1/r)
KE =  5.0 x 107 J - 6.67 x 10-11 x 5.0 x 1023 x 10 x (1 / 3.1 x 106 - 1/4.0 x 106)
KE = 2.58 x 107 J
b) For maximum distance achieved, the final kinetic energy of the probe is zero.
Again using the conservation of energy equation
KEf = KEi - G M m (1/ R - 1/r)
KEi = KEf + G M m (1/ R - 1/r)
KEi = 0 + 6.67 x 10-11 x 5.0 x 1023 x 10 x (1 / 3.1 x 106 - 1/ 8.0x 106)
KEi = 6.589 x 107 J


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