1. Suppose a rocket is fired vertically upward from the surface of Earth with on

Question

1. Suppose a rocket is fired vertically upward from the surface of Earth with one-half of the escape speed. How far from the center of Earth will it reach before it begins to fall back? (Let g = 9.8 m/s2 and RE = 6370 km.)

I need to know how to do them, not just the answer, thank you

Suppose a rocket is fired vertically upward from the surface of Earth with one-half of the escape speed. How far from the center of Earth will it reach before it begins to fall back? (Let g = 9.8 m/s2 and RE = 6370 km.)

Explanation / Answer

Let us use conservation of energy.

-GMm/R + 1/2 mv^2 = 0 gives us the value of escape speed v. Using the fact that g= GM/R^2, you can find out that v^2 = 2gR.

Now we reduce the speed by half. So, v'^2 = 2gR(1/2) ^2 = 1/2gR.

Plug this in back into the equation for conservation of energy. It reads as

-GMm/R + 1/2 mv'^2 = -GMm / R',

where R' is the value we want to know, since that is when the total kinetic energy is 0, hence all the energy is transformed into potential energy. You will find R' = 4/3 R, which gives answer 8.5 * 103 km

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