Let’s imagine that the Earth’s orbital speed suddenly went to zero, and the Eart

Question

Let’s imagine that the Earth’s orbital speed suddenly went to zero, and the Earth began to “fall” directly toward the Sun. What would be the translational speed of the Earth (in km/s) when the Earth had reached the orbit of Mercury, 5.79 x 10^10 m? The mass of the Sun is 1.99 x 10^30 kg, the mass of the Earth is 5.97 x 10^24 kg, and the initial separation distance between the Earth and the Sun is 1.5 x 10^11 m. For simplicity, let’s assume that the Earth and the Sun can be treated as point particles during this entire process and that the Earth-Sun system can be considered isolated.

Explanation / Answer

using conservation of energy:

Ei = Ef

KEi + PEi = KEf + PEf

0 + (-G*Ms*Me/x0) = 0.5*Me*v^2 + (-G*Ms*Me/x)

Me*v^2 = 2*G*Ms*Me*(1/x - 1/x0)

v = sqrt [2*G*Ms*(1/x - 1/x0)]

v = sqrt [2*6.67*10^-11*1.99*10^30*(1/(5.79*10^10) - 1/(1.5*10^11))]

v = 53057.81 m/sec = 53.057 km/sec

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