Two bodies of masses m and 4 m are moving with equal K.E. The ratio of their linear momentum is

Options

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

Explanation

To solve the problem, we need to find the ratio of the linear momentum of two bodies with masses \( m \) and \( 4m \) that are moving with equal kinetic energy.

### Step-by-Step Solution:

1. **Understand the relationship between 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 given by:
\[ P = mv \]

2. **Set up the equations for both bodies**:
Let the kinetic energy of both bodies be \( KE \). For the first body with mass \( m \):
\[ KE_1 = \frac{1}{2} m v_1^2 \]
For the second body with mass \( 4m \):
\[ KE_2 = \frac{1}{2} (4m) v_2^2 \]

3. **Since both bodies have equal kinetic energy**:
\[ KE_1 = KE_2 \]
Therefore,
\[ \frac{1}{2} m v_1^2 = \frac{1}{2} (4m) v_2^2 \]

4. **Simplify the equation**:
Cancel \( \frac{1}{2} \) and \( m \) (assuming \( m \neq 0 \)):
\[ v_1^2 = 4 v_2^2 \]

5. **Relate the velocities**:
Taking the square root of both sides gives:
\[ v_1 = 2 v_2 \]

6. **Find the momentum for both bodies**:
The momentum of the first body is:
\[ P_1 = mv_1 = m(2v_2) = 2mv_2 \]
The momentum of the second body is:
\[ P_2 = (4m)v_2 = 4mv_2 \]

7. **Calculate the ratio of their momenta**:
The ratio of the momenta \( P_1 \) to \( P_2 \) is:
\[ \frac{P_1}{P_2} = \frac{2mv_2}{4mv_2} = \frac{2}{4} = \frac{1}{2} \]

### Final Answer:
The ratio of their linear momentum is \( 1:2 \).