Two masses of $1\,\mathrm{g}$ and $4\,\mathrm{g}$ are moving with equal kinetic energies. The ratio of the magnitudes of their linear momenta is

Step 1: Identify the given values.

The masses of the two objects and their kinetic energies are given.

$$m_1 = 1\,\text{g}$$ $$m_2 = 4\,\text{g}$$ $$K_1 = K_2$$

Step 2: Relate kinetic energy and linear momentum.

The relationship between kinetic energy $K$, mass $m$, and linear momentum $p$ is:

$$K = \frac{p^2}{2m}$$

Rearranging this formula for momentum $p$ yields:

$$p = \sqrt{2mK}$$

Step 3: Determine the ratio of the linear momenta.

Since the kinetic energies are equal ($K_1 = K_2$), the momentum is directly proportional to the square root of the mass.

$$p \propto \sqrt{m}$$

Therefore, the ratio of their momenta is:

$$\frac{p_1}{p_2} = \sqrt{\frac{m_1}{m_2}}$$

Step 4: Substitute the given values to find the ratio.

Substitute $m_1 = 1\,\text{g}$ and $m_2 = 4\,\text{g}$ into the ratio.

$$\frac{p_1}{p_2} = \sqrt{\frac{1}{4}}$$ $$\frac{p_1}{p_2} = \frac{1}{2}$$ $$p_1 : p_2 = 1 : 2$$

Answer: The ratio of the magnitudes of their linear momenta is $1 : 2$.