> Q. The escape velocity from a planet is $’V_e’.$ A tunnel is dug along along a diameter of the planet and a small body is dropped into it. The speed of the body at the centre of the planet will be – LIVE ANSWER TODAY

Q. The escape velocity from a planet is $’V_e’.$ A tunnel is dug along along a diameter of the planet and a small body is dropped into it. The speed of the body at the centre of the planet will be

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The escape velocity from a planet is $’V_e’.$ A tunnel is dug along along a diameter of the planet and a small body is dropped into it. The speed of the body at the centre of the planet will be

Solution:

$V_e = sqrt{2gR}$

According to the law of conservation of energy

$U_e + frac{1}{2} mV^2 = U_s Rightarrow frac{1}{2} mV^2 = U_s – U_c = m(V_s – V_c) $

$Rightarrow frac{1}{2} mV^2 = mleft[ – frac{GM}{R} – left( – frac{3GM}{2R} right) right]$

$Rightarrow frac{1}{2}mV^2 = m left(frac{GM}{2R}right) , Rightarrow V = sqrt{frac{GM}{R}} = frac{V}{sqrt{2}} $

[$therefore$V =$sqrt{gR}$ ]

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