Understanding Kinetic Energy and Momentum
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass and \( v \) is the velocity of the object. The momentum (p) of an object is defined as:
\[ p = mv \]
Given two bodies with masses \( m \) and \( 4m \) and their kinetic energies are in the ratio of 1:2, we can analyze their momentum.

Step 1: Define Kinetic Energies
- Let the kinetic energy of mass \( m \) be \( KE_1 = \frac{1}{2} m v_1^2 \)
- Let the kinetic energy of mass \( 4m \) be \( KE_2 = \frac{1}{2} (4m) v_2^2 \)
Since the ratio of their kinetic energies is given as 1:2:
\[ \frac{KE_1}{KE_2} = \frac{1}{2} \]
This implies:
\[ \frac{\frac{1}{2} m v_1^2}{\frac{1}{2} (4m) v_2^2} = \frac{1}{2} \]

Step 2: Simplifying the Ratio
- Cancelling \(\frac{1}{2}\) and \(m\) (as long as \(m \neq 0\)):
\[ \frac{v_1^2}{4 v_2^2} = \frac{1}{2} \]
- Rearranging gives:
\[ v_1^2 = 2 v_2^2 \]
- Taking square roots results in:
\[ v_1 = v_2 \sqrt{2} \]

Step 3: Finding the Momentum Ratio
- Now, calculate the momentum for both bodies:
\[ p_1 = mv_1 = m(v_2 \sqrt{2}) = mv_2 \sqrt{2} \]
\[ p_2 = (4m)v_2 = 4mv_2 \]
- The ratio of their momenta is:
\[ \frac{p_1}{p_2} = \frac{mv_2 \sqrt{2}}{4mv_2} = \frac{\sqrt{2}}{4} \]

Conclusion
Thus, the ratio of their momenta \( p_1 : p_2 \) is:
\[ \frac{\sqrt{2}}{4} : 1 \]
This means the momentum of the two bodies is in the ratio \( \sqrt{2} : 4 \).