Question 27: Gravitation

Question

If the earth has a mass nine times and radius four times that of planet X, the ratio of the maximum speed required by a rocket to pull out of the gravitational force of planet $$\mathrm{X}$$ to that of the earth is

Accepted Answer

To find the ratio of the maximum speed required by a rocket to escape the gravitational force of planet $$\mathrm{X}$$ to that of Earth, we must consider the escape velocity formula:
$$v_e = \sqrt{\frac{2GM}{R}}$$
where:
$$v_e$$ is the escape velocity
$$G$$ is the gravitational constant
$$M$$ is the mass of the planet
$$R$$ is the radius of the planet
Given:
The mass of Earth ($$M_{	ext{Earth}}$$) is nine times the mass of planet X ($$M_X$$),
The radius of Earth ($$R_{	ext{Earth}}$$) is four times the radius of planet X ($$R_X$$).
So, we have:
$$M_{	ext{Earth}} = 9M_X$$
$$R_{	ext{Earth}} = 4R_X$$
The escape velocity for planet X is:
$$v_{e,X} = \sqrt{\frac{2GM_X}{R_X}}$$
The escape velocity for Earth is:
$$v_{e, 	ext{Earth}} = \sqrt{\frac{2GM_{	ext{Earth}}}{R_{	ext{Earth}}}}$$
Substitute $$M_{	ext{Earth}} = 9M_X$$ and $$R_{	ext{Earth}} = 4R_X$$:
$$v_{e, 	ext{Earth}} = \sqrt{\frac{2G(9M_X)}{4R_X}} = \sqrt{\frac{18GM_X}{4R_X}} = \sqrt{\frac{9GM_X}{2R_X}} = 3\sqrt{\frac{GM_X}{2R_X}}$$
However, note that:
$$v_{e,X} = \sqrt{\frac{2GM_X}{R_X}}$$
So, the ratio of the escape velocity of planet X to Earth is:
$$\frac{v_{e,X}}{v_{e, 	ext{Earth}}} = \frac{\sqrt{\frac{2GM_X}{R_X}}}{3\sqrt{\frac{GM_X}{2R_X}}}$$
$$= \frac{\sqrt{\frac{2GM_X}{R_X}}}{3 \cdot \sqrt{\frac{GM_X}{2R_X}}}$$
$$= \frac{1}{3} \cdot \sqrt{\frac{2GM_X}{R_X} \cdot \frac{2R_X}{GM_X}}$$
$$= \frac{1}{3} \sqrt{4}$$
$$= \frac{1}{3} \cdot 2$$
$$= \frac{2}{3}$$
Therefore, the ratio of the maximum speed required by a rocket to escape the gravitational force of planet $$\mathrm{X}$$ to that of Earth is:
$$\boxed{\frac{2}{3}}$$

Answer

$$\frac{2}{3}$$

Answer

$$\frac{9}{4}$$

Answer

$$\frac{3}{2}$$

Answer

$$\frac{4}{9}$$

Gravitation

If the earth has a mass nine times and radius four times that of planet X, the ratio of the maximum speed required by a rocket to pull out of the gravitational force of planet $\mathrm{X}$ to that of the earth is

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