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Chapter 6: Problem 761

If \(\mathrm{V}_{\mathrm{e}}\) and \(\mathrm{V}_{\mathrm{o}}\) are represent the escape velocity and orbital velocity of satellite corresponding to a circular orbit of -adius \(\mathrm{r}\), then A) \(\mathrm{V}_{\mathrm{e}}=\mathrm{V}_{\mathrm{o}}\) (B) \(\sqrt{2} \mathrm{~V}_{\mathrm{o}}=\mathrm{V}_{\mathrm{e}}\) C) \(\mathrm{V}_{\mathrm{e}}=\left(\mathrm{V}_{\mathrm{O}} / \sqrt{2}\right)\) (D) \(\mathrm{V}_{\mathrm{e}}\) and \(\mathrm{V}_{\mathrm{o}}\) are not related

Short Answer

Expert verified
The short answer is: \(V_e = \sqrt{2}V_o\). Option (B) is correct.

Step by step solution

01

1. Find the formula for escape velocity

The escape velocity is the minimum velocity an object needs to break free from a massive body's gravitational pull. The formula for escape velocity is given by: \(V_e = \sqrt{ \dfrac{2GM}{r} }\) where G is the gravitational constant, M is the mass of the massive body (like Earth), and r is the distance from the center of the massive body (radius of the orbit).
02

2. Find the formula for orbital velocity

The orbital velocity is the velocity needed for an object to stay in a stable, circular orbit around a massive body. The formula for orbital velocity is given by: \(V_o = \sqrt{ \dfrac{GM}{r} }\) where G is the gravitational constant, M is the mass of the massive body (like Earth), and r is the distance from the center of the massive body (radius of the orbit).
03

3. Compare escape and orbital velocities

Now let's compare the formulas for escape and orbital velocities: \(V_e = \sqrt{\dfrac{2GM}{r}} \) and \( V_o = \sqrt{\dfrac{GM}{r}} \) To find the relationship between Ve and Vo, we can try to express Ve in terms of Vo. Let's square both Ve and Vo: \(V_e^2 = \dfrac{2GM}{r}\) and \(V_o^2 = \dfrac{GM}{r}\) Now, let's divide Ve^2 by Vo^2: \(\dfrac{V_e^2}{V_o^2} = \dfrac{\dfrac{2GM}{r}}{\dfrac{GM}{r}}\) This simplifies to: \(\dfrac{V_e^2}{V_o^2} = 2\) Now take the square root of both sides: \( \dfrac{V_e}{V_o} = \sqrt{2}\) From this relationship, we can see that: \(V_e = \sqrt{2}V_o\) So, the correct option is (B) \(\sqrt{2}V_o = V_e\).

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