Question 105: Gravitation

Question

A space ship of mass $$2 	imes 10^{4} \mathrm{~kg}$$ is launched into a circular orbit close to the earth surface. The additional velocity to be imparted to the space ship in the orbit to overcome the gravitational pull will be (if $$g=10 \mathrm{~m} / \mathrm{s}^{2}$$ and radius of earth $$=6400 \mathrm{~km}$$ ):

Accepted Answer

To find the additional velocity required to overcome the gravitational pull and launch the spaceship into orbit, we first need to find the orbital velocity. The formula for orbital velocity (v) is given by:
$$v = \sqrt{\frac{GM}{r}}$$
where G is the gravitational constant, M is the mass of Earth, and r is the distance from the center of the Earth to the spaceship (which is the sum of the Earth's radius and the altitude of the spaceship's orbit).
In this problem, the spaceship is orbiting close to Earth's surface, so we can approximate r as the Earth's radius. Given that g = 10 m/sÃÂ² and Earth's radius R = 6400 km, we can relate the gravitational constant G and the mass of Earth M through the formula:
$$g = \frac{GM}{R^2}$$
Now, we can find the orbital velocity:
$$v = \sqrt{\frac{gR^2}{R}} = \sqrt{gR}$$
Converting the Earth's radius to meters:
$$R = 6400 	imes 10^3 \mathrm{~m}$$
Plugging in the values:
$$v = \sqrt{10 	imes 6400 	imes 10^3} = 8 	imes 10^3 \mathrm{~m/s}$$
Now, we need to find the additional velocity required to overcome the gravitational pull. To do this, we can use the formula for escape velocity:
$$v	ext{escape} = \sqrt{2} 	imes v	ext{orbital}$$
Finding the additional velocity required:
$$\Delta v = v	ext{escape} - v	ext{orbital} = (\sqrt{2} - 1) 	imes v_	ext{orbital}$$
Plugging in the values:
$$\Delta v = (\sqrt{2} - 1) 	imes 8 	imes 10^3 \mathrm{~m/s}$$
Converting the velocity to km/s:
$$\Delta v = (\sqrt{2} - 1) 	imes 8 \mathrm{~km/s}$$
Thus, the additional velocity required to overcome the gravitational pull and launch the spaceship into orbit is:
$$8(\sqrt{2}-1) \mathrm{~km/s}$$

Answer

$$7.9(\sqrt{2}-1) \mathrm{km} / \mathrm{s}$$

Answer

$$11.2(\sqrt{2}-1) \mathrm{km} / \mathrm{s}$$

Answer

$$7.4(\sqrt{2}-1) \mathrm{km} / \mathrm{s}$$

Answer

$$8(\sqrt{2}-1) \mathrm{km} / \mathrm{s}$$

Gravitation

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