MHT CET · Physics · Gravitation
A body (mass \(\mathrm{m}\) ) starts its motion from rest from a point distant \(R_0\left(R_0>R\right)\) from the centre of the earth. The velocity acquired by the body when it reaches the surface of earth will be ( \(\mathrm{G}=\) universal constant of gravitation, \(\mathrm{M}=\) mass of earth, \(\mathrm{R}\) = radius of earth)
- A \(2 \mathrm{GM}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)\)
- B \(\left[2 \mathrm{GM}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)\right]^{\frac{1}{2}}\)
- C \(\mathrm{GM}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)\)
- D \(2 \mathrm{GM}\left[\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)\right]^{\frac{1}{2}}\)
Answer & Solution
Correct Answer
(B) \(\left[2 \mathrm{GM}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)\right]^{\frac{1}{2}}\)
Step-by-step Solution
Detailed explanation
According to law of conservation of energy,
\(\begin{aligned}
& \quad \frac{1}{2} \mathrm{mv}^2=-\frac{\mathrm{GMm}}{\mathrm{R}_0}-\left(\frac{\mathrm{GMm}}{\mathrm{R}}\right)=\mathrm{GMm}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right) \\
& \therefore \quad \mathrm{v}^2=2 \mathrm{GM}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)
\end{aligned}\)
\(\therefore \quad\) The velocity acquired by the body when it reaches the surface of earth is:
\(\mathrm{V}=\left[2 \mathrm{GM}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)\right]^{\frac{1}{2}}\)
\(\begin{aligned}
& \quad \frac{1}{2} \mathrm{mv}^2=-\frac{\mathrm{GMm}}{\mathrm{R}_0}-\left(\frac{\mathrm{GMm}}{\mathrm{R}}\right)=\mathrm{GMm}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right) \\
& \therefore \quad \mathrm{v}^2=2 \mathrm{GM}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)
\end{aligned}\)
\(\therefore \quad\) The velocity acquired by the body when it reaches the surface of earth is:
\(\mathrm{V}=\left[2 \mathrm{GM}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)\right]^{\frac{1}{2}}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Physics
- A thin hollow prism of refracting angle filled with water gives a deviation of The refractive index of water isMHT CET 2019 Easy
- Two open organ pipes of fundamental frequencies \(\mathrm{n}_{1}\) and \(\mathrm{n}_{2}\) are joined in series. The fundamental frequency of the new pipe isMHT CET 2020 Medium
- A body of mass \(0.04 \mathrm{~kg}\) excutes simple harmonic motion (SHM) about \(\mathrm{x}=0\) under the influence of force \(\mathrm{F}\) as shown in graph. The period of
MHT CET 2023 Hard - For a particle performing S.H.M. the equation \(\left(\frac{\mathrm{d}^2 \mathrm{x}}{\mathrm{dt}^2}\right)+\alpha \mathrm{x}=0\). Then the time period of the motion will beMHT CET 2022 Medium
- A particle is moving in U.C.M. with speed \(V\) and radius \(R\). The angular acceleration of the particle isMHT CET 2022 Medium
- A big water drop is formed by the combination of ' \(n\) ' small water droplets of equal radii. The ratio of the surface energy of ' \(n\) ' droplets to the surface energy of the big drop isMHT CET 2024 Medium
More PYQs from MHT CET
- A stone is dropped into a pond. Waves in the form of circles are generated and radius of outermostripple increases at the rate of Then area increased after seconds is ________MHT CET 2019 Medium
- If the position vectors of the vertices \(A, B, C\) of a tringle \(A B C\) are \(4 \hat{\imath}+7 j+8 \hat{k}, 2 \hat{\imath}+3 \hat{\jmath}+4 \hat{k}\) and \(2 \hat{\imath}+5 \hat{\jmath}+7 \hat{k}\) respectively, then the position vector of the point where bisector of angle A meets \(B C\) isMHT CET 2020 Medium
- If \(y=\tan ^{-1}\left(\frac{5 x+1}{3-x-6 x^2}\right)\), then \(\frac{d y}{d x}=\)MHT CET 2022 Medium
- If the Vant Hoff factor for solution is the degree of dissociation isMHT CET 2019 Medium
- The mean and the standard deviation of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by p and then reduced by \(q\), where \(p \neq 0\) and \(q \neq 0\). If the new mean and new standard deviation (s.d.) become half of the original values, then q is equal toMHT CET 2024 Hard
- Two sides of a triangle are \(\sqrt{3}+1\) and \(\sqrt{3}-1\) and the included angle is \(60^{\circ}\), then the difference of the remaining angles isMHT CET 2023 Medium