Two bodies of masses 4 kg and 5 kg are moving with equal momentum. Then the ratio of their respective kinetic energies is

Options

A. (a)`4:5`
B. (b)`2:1`
C. (c)`1:3`
D. (d)`5:4`

Answer

To solve the problem, we need to find the ratio of the kinetic energies of two bodies with masses 4 kg and 5 kg that are moving with equal momentum.

### Step-by-Step Solution:

1. **Understanding Momentum**:
The momentum (P) of an object is given by the formula:
\[ P = m \cdot v \]
where \( m \) is the mass and \( v \) is the velocity of the object.

2. **Given Information**:
We have two bodies with masses:
- Mass of body 1, \( m_1 = 4 \, \text{kg} \)
- Mass of body 2, \( m_2 = 5 \, \text{kg} \)
It is given that both bodies have equal momentum, i.e., \( P_1 = P_2 \).

3. **Setting Up the Equation for Momentum**:
For body 1:
\[ P_1 = m_1 \cdot v_1 = 4 \cdot v_1 \]
For body 2:
\[ P_2 = m_2 \cdot v_2 = 5 \cdot v_2 \]
Since \( P_1 = P_2 \):
\[ 4 \cdot v_1 = 5 \cdot v_2 \]

4. **Expressing Velocities**:
From the equation \( 4 \cdot v_1 = 5 \cdot v_2 \), we can express \( v_1 \) in terms of \( v_2 \):
\[ v_1 = \frac{5}{4} v_2 \]

5. **Kinetic Energy Formula**:
The kinetic energy (KE) of an object is given by:
\[ KE = \frac{1}{2} m v^2 \]

6. **Calculating Kinetic Energies**:
- Kinetic energy of body 1:
\[ KE_1 = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} \cdot 4 \cdot v_1^2 \]
- Kinetic energy of body 2:
\[ KE_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} \cdot 5 \cdot v_2^2 \]

7. **Substituting for \( v_1 \)**:
Substitute \( v_1 = \frac{5}{4} v_2 \) into the equation for \( KE_1 \):
\[ KE_1 = \frac{1}{2} \cdot 4 \cdot \left(\frac{5}{4} v_2\right)^2 = \frac{1}{2} \cdot 4 \cdot \frac{25}{16} v_2^2 = \frac{100}{32} v_2^2 = \frac{25}{8} v_2^2 \]

8. **Finding the Ratio of Kinetic Energies**:
Now, we find the ratio of the kinetic energies:
\[ \frac{KE_1}{KE_2} = \frac{\frac{25}{8} v_2^2}{\frac{1}{2} \cdot 5 \cdot v_2^2} = \frac{\frac{25}{8} v_2^2}{\frac{5}{2} v_2^2} \]
The \( v_2^2 \) cancels out:
\[ \frac{KE_1}{KE_2} = \frac{25/8}{5/2} = \frac{25}{8} \cdot \frac{2}{5} = \frac{25 \cdot 2}{8 \cdot 5} = \frac{50}{40} = \frac{5}{4} \]

9. **Final Answer**:
Therefore, the ratio of their respective kinetic energies is:
\[ \frac{KE_1}{KE_2} = \frac{5}{4} \]