Planet $A$ has mass $M$ and radius $R$. Planet $B$ has half the mass and half the radius of Planet $A$. If the escape velocities from the Planets $A$ and $B$ are $v_A$ and $v_B$, respectively, then $\frac{v_{A}}{v_{B}}=\frac{n}{4}$. The value of $n$ is :
- $3$
- $1$
- $2$
- $4$
The Correct Option is D
Solution and Explanation
$V_{A} = \sqrt{\frac{2GM}{R}}$
$V_{B} = \sqrt{\frac{2G\left[M/ 2\right]}{R/ 2}} = \sqrt{\frac{2GM}{R}}$
$\frac{V_{A}}{V_{B}} = 1 = \frac{n}{4} \Rightarrow n = 4$
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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].