Gravitation Question 344
Question: A space vehicle approaching a planet has a speed v, when it is very far from the planet. At that moment tangent of its trajectory would miss the centre of the planet by distance R. If the planet has mass M and radius r, what is the smallest value of R in order that the resulting orbit of the space vehicle will just miss the surface of the planet?
Options:
A) $ [\frac{r}{v}{{\left[ v^{2}+\frac{2GM}{r} \right]}^{\frac{1}{2}}}] $
B) $ [vr\left[ 1+\frac{2GM}{r} \right]] $
C) $ [\frac{r}{v}\left[ v^{2}+\frac{2GM}{r} \right]] $
D) $ [\frac{2GMv}{r}] $
Show Answer
Answer:
Correct Answer: A
Solution:
-
From the principle of conserving angular momentum, we have
$ [MvR=mv’r] $ …(i) [$ [v’=] $ speed when spaceship is just touching the plane] From conserving of energy,
we have $ [\frac{1}{2}mv^{2}=\frac{1}{2}mv{{’}^{2}}-\frac{GMm}{r}] $ -.(ii) Solving Eqs. (i) and (ii), we get $ [R=\frac{r}{v}{{\left[ v^{2}+\frac{2GM}{r} \right]}^{1/2}}] $
