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Standard XII

Physics

Escape Speed

The masses an...

Question

The masses and the radii of the earth and the moon are $M_{1}, R_{1}$ and $M_{2}, R_{2}$ respectively. Their centres are at a distance $d$ apart. Find the minimum speed with which a particle of mass $m$ should be projected from a point midway between the two centres so as to escape it to infinity.

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Solution

The P.E. of the mass at $d / 2$ due to the earth and the moon is
$U = \frac{- G M_{1} m}{d} \times 2 - \frac{G M_{2} m}{d} \times 2$
or $U = \frac{2 G m}{d} (M_{1} + M_{2}) (n u m e r i c a l l y) = \frac{1}{2} m V_{e}^{2} = U$
$V_{e} = 2 \sqrt{[\frac{G^{'}}{d} (M_{1} + M_{2})]}$ .

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The masses and the radii of the earth and the moon are M1,R1 and M2,R2 respectively. Their centres are at a distance d apart. Find the minimum speed with which a particle of mass m should be projected from a point midway between the two centres so as to escape it to infinity.

The P.E. of the mass at d/2 due to the earth and the moon isU=−GM1md×2−GM2md×2or U=2Gmd(M1+M2)(numerically)=12mV2e=UVe=2√[G′d(M1+M2)].

The masses and the radii of the earth and the moon are $M_{1}, R_{1}$ and $M_{2}, R_{2}$ respectively. Their centres are at a distance $d$ apart. Find the minimum speed with which a particle of mass $m$ should be projected from a point midway between the two centres so as to escape it to infinity.

The P.E. of the mass at $d / 2$ due to the earth and the moon is
$U = \frac{- G M_{1} m}{d} \times 2 - \frac{G M_{2} m}{d} \times 2$
or $U = \frac{2 G m}{d} (M_{1} + M_{2}) (n u m e r i c a l l y) = \frac{1}{2} m V_{e}^{2} = U$
$V_{e} = 2 \sqrt{[\frac{G^{'}}{d} (M_{1} + M_{2})]}$ .