A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be:
- 800 km
- 1600 km
- 2133 km
- 4800 km
The Correct Option is A
Solution and Explanation
As per the theory of conservation of energy
\(-\frac{ GM_em }{ R_e} + \frac{1}{2}m\bigg(\frac{1}{3}\sqrt{\frac{2Gm_e}{R_e)}}\bigg)^2 = - \frac{GM_em}{R_e+h}\)
- \(-\frac{GM_em}{R_e} + \frac{GM_em}{9R_e} = -\frac{GM_em}{R_e+h}\)
\(\frac{8}{9R_e} = \frac{1}{R_e+h}\)
Therefore, the maximum height attained by the body will be:
\(⇒ h = \frac{R_e}{8}\)
= \(\frac{6400}{8}\)
= \(800\) \(km\)
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\[ \begin{array}{|l|l|} \hline \text{LIST-I} & \text{LIST-II} \\ \hline \text{A. Gravitational constant} & \text{I. } [LT^{-2}] \\ \hline \text{B. Gravitational potential energy} & \text{II. } [L^2T^{-2}] \\ \hline \text{C. Gravitational potential} & \text{III. } [ML^2T^{-2}] \\ \hline \text{D. Acceleration due to gravity} & \text{IV. } [M^{-1}L^3T^{-2}] \\ \hline \end{array} \]
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Concepts Used:
Gravitation
In mechanics, the universal force of attraction acting between all matter is known as Gravity, also called gravitation, . It is the weakest known force in nature.
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,
- F ∝ (M1M2) . . . . (1)
- (F ∝ 1/r2) . . . . (2)
On combining equations (1) and (2) we get,
F ∝ M1M2/r2
F = G × [M1M2]/r2 . . . . (7)
Or, f(r) = GM1M2/r2
The dimension formula of G is [M-1L3T-2].