A particle released at a large distance from a planet reaches the… - Vidyadip

MCQ

A particle released at a large distance from a planet reaches the planet only under gravitational attraction and passes through a smooth tunnel through its centre. If $v_e$ is the escape velocity of a body at the planet, then the particle's speed at the centre of the planet is

✓
$\sqrt{1.5} v _{ e }$
B
$v _{ e }$
C
$1.5 v _{ e }$
D
$2 v _{ e }$

✓

Answer

Correct option: A.

$\sqrt{1.5} v _{ e }$

a
Taking the potential at a large distance from the planet as zero, the potential at the centre of the planet $=-3 GM / 2 R$

$K_A+U_A=K_O+U_O$

$0+0=\frac{1}{2} v^2+m V_O$

$=\frac{1}{2} m v^2+m\left(-\frac{3 G M}{2 R}\right)$

$v=\sqrt{\frac{3 G M}{R}}=\sqrt{\frac{3}{2}} \cdot \sqrt{\frac{2 G M}{R}}$

$v=\sqrt{1.5}{ v_e}$

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