Question

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,

• A. $27:64$
• B. $1:1$
• C. $27:256$
• D. $9:64$

Answer 1

Answer: (C)

$27:256$

Given,

Ratio of radii, $r=1:2$

Ratio of velocity, $v=3:4$

Ratio of density, $d=3:2$

Let the kinetic energy of the spheres be $K{E}_1$ and $K{E}_2$

Then,

$\frac{K{E}_1}{K{E}_2}=\frac{\frac{1}{2}{m}_1{v}_1^2}{\frac{1}{2}{m}_2{v}_2^2}=\frac{\frac{1}{2}(d_1\times V_1)v_1^2}{\frac{1}{2}(d_2\times V_2)v_2^2}=\frac{\frac{1}{2}(d_1\times \frac{4\pi r_1^3}{3})v_1^2}{\frac{1}{2}(d_2\times \frac{4\pi r_2^3}{3}{v}_2^2)}=\frac{d_1}{d_2}\times (\frac{r_1}{r_2}{)}^3\times (\frac{v_1}{v_2}{)}^2$

$=\frac{3}{2}\times (\frac{1}{2}{)}^3\times (\frac{3}{4}{)}^2=\frac{27}{256}$

So the ratio of their kinetic energies is $27:256$