Two bodies of mass `1 kg and 2 kg` have equal momentum. The ratio of their kinetic energies is:

Options

A. `1:3`
B. `3:1`
C. `2:1`
D. `4:1`

Answer

To solve the problem, we need to find the ratio of the kinetic energies of two bodies with masses 1 kg and 2 kg that have equal momentum. Let's break this down step by step.

### Step 1: Understand the relationship between momentum and kinetic energy
Momentum (p) is defined as:
\[ p = mv \]
where \( m \) is the mass and \( v \) is the velocity of the body.

Kinetic energy (KE) is defined as:
\[ KE = \frac{1}{2} mv^2 \]

### Step 2: Set up the equations for both bodies
Let:
- Mass of body 1 (m1) = 1 kg
- Mass of body 2 (m2) = 2 kg

Since both bodies have equal momentum, we can write:
\[ p_1 = p_2 \]
This implies:
\[ m_1 v_1 = m_2 v_2 \]
Substituting the masses:
\[ 1 \cdot v_1 = 2 \cdot v_2 \]
From this, we can express \( v_1 \) in terms of \( v_2 \):
\[ v_1 = 2 v_2 \]

### Step 3: Calculate the kinetic energies of both bodies
Now, we can calculate the kinetic energies of both bodies.

For body 1:
\[ KE_1 = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} \cdot 1 \cdot (2 v_2)^2 \]
\[ KE_1 = \frac{1}{2} \cdot 1 \cdot 4 v_2^2 = 2 v_2^2 \]

For body 2:
\[ KE_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} \cdot 2 \cdot v_2^2 \]
\[ KE_2 = 1 v_2^2 \]

### Step 4: Find the ratio of kinetic energies
Now we can find the ratio of their kinetic energies:
\[ \text{Ratio} = \frac{KE_1}{KE_2} = \frac{2 v_2^2}{1 v_2^2} = \frac{2}{1} \]

### Conclusion
Thus, the ratio of the kinetic energies of the two bodies is:
\[ \text{Ratio} = 2:1 \]