Question:
The mass of a spaceship is $1000 \,kg$. It is to be launched from the earth's surface out into free space. The value of $g$ and $R$ (radius of earth) are $10\, m/s^2$ and $6400\, km$ respectively. The required energy for this work will be :
The mass of a spaceship is $1000 \,kg$. It is to be launched from the earth's surface out into free space. The value of $g$ and $R$ (radius of earth) are $10\, m/s^2$ and $6400\, km$ respectively. The required energy for this work will be :
Updated On: Apr 7, 2024
- $6.4 ??10^{11}$ Joules
- $6.4 ??10^{8}$ Joules
- $6.4 ??10^{9}$ Joules
- $6.4 ??10^{10}$ Joules
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The Correct Option is D
Solution and Explanation
$W=0-\left(-\frac{GMm}{R}\right)=\frac{GMm}{R}$
$=gR^{2}\times \frac{m}{R}=mgR$
$= 1000 ??10 ??6400 ??10^{3}$
$= 64 ??10^{9}\, J$
$= 64 ??10^{10}$
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Concepts Used:
Newtons Law of Gravitation
Gravitational Force
Gravitational force is a central force that depends only on the position of the test mass from the source mass and always acts along the line joining the centers of the two masses.
Newton’s Law of Gravitation:
According to Newton’s law of gravitation, “Every particle in the universe attracts every other particle with a force whose magnitude is,
- Directly proportional to the product of their masses i.e. F ∝ (M1M2) . . . . (1)
- Inversely proportional to the square of the distance between their center i.e. (F ∝ 1/r2) . . . . (2)
By combining equations (1) and (2) we get,
F ∝ M1M2/r2
F = G × [M1M2]/r2 . . . . (7)
Or, f(r) = GM1M2/r2 [f(r)is a variable, Non-contact, and conservative force]