A spaceship of mass is launched into a circular orbit close to the Earth’s surface. The additional velocity to be imparted to the spaceship in the orbit to overcome the gravitational pull will be (if and radius of Earth = 6400 km) [2023

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Q.
A spaceship of mass $2 \times 10^{4} kg$ is launched into a circular orbit close to the Earth’s surface. The additional velocity to be imparted to the spaceship in the orbit to overcome the gravitational pull will be (if $g = 10 {m/s}^{2}$ and radius of Earth = 6400 km) [2023]

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

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

3 $8 (\sqrt{2} - 1) km/s$

4 $7.4 (\sqrt{2} - 1) km/s$

Ans.
(3)

$v_{orbit} = \sqrt{\frac{G M}{R}} = \sqrt{g R}$

$v_{escape} = \sqrt{\frac{2 G M}{R}} = \sqrt{2 g R}$

$Δ v = (\sqrt{2} - 1) \sqrt{g R} = 8 (\sqrt{2} - 1) km/s$

Solution

Q.
A spaceship of mass $2 \times 10^{4} kg$ is launched into a circular orbit close to the Earth’s surface. The additional velocity to be imparted to the spaceship in the orbit to overcome the gravitational pull will be (if $g = 10 {m/s}^{2}$ and radius of Earth = 6400 km) [2023]

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

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

3 $8 (\sqrt{2} - 1) km/s$

4 $7.4 (\sqrt{2} - 1) km/s$

Ans.
(3)

$v_{orbit} = \sqrt{\frac{G M}{R}} = \sqrt{g R}$

$v_{escape} = \sqrt{\frac{2 G M}{R}} = \sqrt{2 g R}$

$Δ v = (\sqrt{2} - 1) \sqrt{g R} = 8 (\sqrt{2} - 1) km/s$

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Solution

Q. A spaceship of mass 2×104 kg is launched into a circular orbit close to the Earth’s surface. The additional velocity to be imparted to the spaceship in the orbit to overcome the gravitational pull will be (if g=10 m/s2 and radius of Earth = 6400 km) [2023]

1 11.2(2-1) km/s

2 7.9(2-1) km/s

3 8(2-1) km/s

4 7.4(2-1) km/s

Ans. (3) vorbit=GMR=gR vescape=2GMR=2gR Δv=(2-1)gR=8(2-1) km/s

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Q.
A spaceship of mass $2 \times 10^{4} kg$ is launched into a circular orbit close to the Earth’s surface. The additional velocity to be imparted to the spaceship in the orbit to overcome the gravitational pull will be (if $g = 10 {m/s}^{2}$ and radius of Earth = 6400 km) [2023]

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

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

3 $8 (\sqrt{2} - 1) km/s$

4 $7.4 (\sqrt{2} - 1) km/s$

Ans.
(3)

$v_{orbit} = \sqrt{\frac{G M}{R}} = \sqrt{g R}$

$v_{escape} = \sqrt{\frac{2 G M}{R}} = \sqrt{2 g R}$

$Δ v = (\sqrt{2} - 1) \sqrt{g R} = 8 (\sqrt{2} - 1) km/s$