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Error and Uncertainty

Two spherical...

Question

Two spherical planets $P$ and $Q$ have the same uniform density $ρ$ , masses $M_{P}$ and $M_{Q}$ , and surface areas $A$ and $4 A$ , respectively. $A$ spherical planet $R$ also has uniform density $ρ$ and its mass is $(M_{P} + M_{Q})$ . The escape velocities from the planets $P, Q$ and $R$ , are $V_{P},$ $V_{Q}$ and $V_{R}$ , respectively. Then

A

V_{Q} > V_{R} > V_{P}

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B

V_{R} > V_{Q} > V_{P}

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C

\frac{V_{P}}{V_{P}} = \frac{1}{2}

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D

\frac{V_{R}}{V_{P}} = 3

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Solution

The correct option is C $\frac{V_{P}}{V_{P}} = \frac{1}{2}$
As area of $Q$ is $4$ times area of $P$ so

radius of $Q = 2$ times radius of $P$

$R_{Q} = 2 R_{P}$

Also, $M_{Q} = 8 M_{P}$

So $M_{R} = M_{Q} + M_{P} = 9 M_{P}$

and $R_{R} = 9^{\frac{1}{3}} R_{P}$

Now, $V_{e} = \sqrt{\frac{G M}{R}}$

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