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Chapter 6: Problem 742

There are two planets, the ratio of radius of two planets is \(\mathrm{k}\) but the acceleration due to gravity of both planets are \(\mathrm{g}\) what will be the ratio of their escape velocity. (A) \((\mathrm{kg})^{1 / 2}\) (B) \((\mathrm{kg})^{-1 / 2}\) (C) \((\mathrm{kg})^{2}\) (D) \((\mathrm{kg})^{-2}\)

Short Answer

Expert verified
The short answer is: (A) \((kg)^{1/2}\).

Step by step solution

01

Write down the escape velocity formula

The escape velocity formula is given by: \[v_e = \sqrt{\frac{2GM}{R}}\] where \(v_e\) is the escape velocity, \(G\) is the gravitational constant, \(M\) is the mass of the planet, and \(R\) is the radius of the planet.
02

Find the ratio of masses of the planets

We are given that the ratio of the radii of the planets is \(k\). We know that the acceleration due to gravity on both planets is the same, so we can write: \[g = \frac{GM_1}{R_1^2} = \frac{GM_2}{R_2^2}\] Dividing the equations and using the given ratio of radii, we get: \[\frac{M_1}{M_2} = \frac{R_1^2}{R_2^2} = k^2\]
03

Write down the escape velocities for both planets

Using the escape velocity formula and the ratio of masses found in the previous step, we can write the escape velocities for both planets as: \[v_{e1} = \sqrt{\frac{2GM_1}{R_1}}\] \[v_{e2} = \sqrt{\frac{2GM_2}{R_2}}\]
04

Find the ratio of escape velocities

Now, we will find the ratio of escape velocities by dividing the equations: \[\frac{v_{e1}}{v_{e2}} = \frac{\sqrt{\frac{2GM_1}{R_1}}}{\sqrt{\frac{2GM_2}{R_2}}}\] Using the ratios found in Step 2, we can simplify this expression: \[\frac{v_{e1}}{v_{e2}} = \sqrt{\frac{M_1}{M_2} \cdot \frac{R_2}{R_1}} = \sqrt{\frac{k^2}{k}} = \sqrt{k}\]
05

Identify the correct answer

Comparing our final answer to the provided options, we find that the correct answer is: (A) \((kg)^{1/2}\).

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As astronaut orbiting the earth in a circular orbit \(120 \mathrm{~km}\) above the surface of earth, gently drops a spoon out of space-ship. The spoon will (A) Fall vertically down to the earth (B) move towards the moon (C) Will move along with space-ship (D) Will move in an irregular wav then fall down to earth

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