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Two spheres whose radii are in the ratio 1 : 2 are moving with their velocities in the ratio 3 : 4. If their densities are in the ratio 3 : 2, the ratio of their kinetic energies is,

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9 : 64

27 : 256

1 : 1

27 : 64

answer is C.

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Detailed Solution

$K E = \frac{1}{2} m v^{2} = \frac{1}{2} (\frac{4}{3} π r^{3} ρ) v^{2} = \frac{2 π}{3} r^{3} ρ v^{2}$

$\frac{K E_{1}}{K E_{2}} = {(\frac{r_{1}}{r_{2}})}^{3} \frac{ρ_{1}}{ρ_{2}} {(\frac{v_{1}}{v_{2}})}^{2} = (\frac{1}{8}) (\frac{3}{2}) (\frac{9}{16}) = \frac{27}{256}$

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