First slide

Gravitaional potential energy

Question

A particle falls towards the earth from infinity. The velocity with which it reaches earth’s surface (g → acceleration due to gravity on earth’s surface and R is the radius of the earth)

Moderate

A

2gR
B

$\large \sqrt {2gR}$
C

$\large \sqrt {gR}$
D

R/g

Solution

Escape velocity $\large {V_e} = \sqrt {\frac{{2GM}}{R}} = \sqrt {2gR}$
Total energy is conserved. $\large \frac 12mV^2+\left ( -\frac {GMm}{R} \right )=0$ $\large \frac 12mV^2+\left ( -\frac {GMm}{R} \right )=0\Rightarrow V=\sqrt \frac {2GM}{R}$

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