Two masses of 1g and 9g are moving with equal kinetic energies. The ratio of the magnitudes of their respective linear momenta is

Correct Option: 1:3

To find the ratio of the linear momenta of the two masses, we can establish a relationship between kinetic energy, mass, and momentum.

The kinetic energy ($K$) of an object of mass $m$ moving with velocity $v$ is given by:

$K=\frac{1}{2}m{v}^2$

The magnitude of linear momentum ($p$) is given by:

$p=mv$

We can rewrite the velocity as $v=\frac{p}{m}$ and substitute it into the kinetic energy formula:

$K=\frac{1}{2}m{\left(\frac{p}{m}\right)}^2=\frac{p^2}{2m}$

Solving for momentum ($p$), we get:

$p=\sqrt{2mK}$

Let the two masses be $m_1=1\text{g}$ and $m_2=9\text{g}$. Since both masses have equal kinetic energies ($K_1=K_2=K$), the ratio of their momenta is:

$\frac{p_1}{p_2}=\frac{\sqrt{2m_{1}K}}{\sqrt{2m_{2}K}}=\sqrt{\frac{m_1}{m_2}}$

Substituting the given values of the masses into this equation:

$\frac{p_1}{p_2}=\sqrt{\frac{1}{9}}=\frac{1}{3}$

Thus, the ratio of the magnitudes of their respective linear momenta is 1:3.