Two masses of 1 g and 4g are moving with equal linear momenta. The ratio of their kinetic energies is :

Options

A. (a) `4:1`
B. (b) `sqrt2:1`
C. (c) `1:2`
D. (d) `1:16`

Explanation

To solve the problem of finding the ratio of kinetic energies of two masses (1 g and 4 g) moving with equal linear momenta, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the given information**:
- Mass 1 (m1) = 1 g = 0.001 kg (convert grams to kilograms for standard SI units)
- Mass 2 (m2) = 4 g = 0.004 kg
- Both masses are moving with equal linear momentum.

2. **Define linear momentum**:
- Linear momentum (P) is given by the formula:
\[ P = m \cdot v \]
- For mass 1, the momentum is:
\[ P_1 = m_1 \cdot v_1 \]
- For mass 2, the momentum is:
\[ P_2 = m_2 \cdot v_2 \]

3. **Set the momenta equal**:
Since the momenta are equal:
\[ P_1 = P_2 \]
This gives us:
\[ m_1 \cdot v_1 = m_2 \cdot v_2 \]

4. **Substitute the values of mass**:
\[ 0.001 \cdot v_1 = 0.004 \cdot v_2 \]

5. **Express \(v_2\) in terms of \(v_1\)**:
Rearranging the equation gives:
\[ v_2 = \frac{0.001}{0.004} \cdot v_1 = \frac{1}{4} v_1 \]

6. **Calculate the kinetic energies**:
- Kinetic energy (K.E) is given by the formula:
\[ K.E = \frac{1}{2} m v^2 \]
- For mass 1:
\[ K.E_1 = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} \cdot 0.001 \cdot v_1^2 \]
- For mass 2:
\[ K.E_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} \cdot 0.004 \cdot v_2^2 \]
- Substitute \(v_2\) from step 5 into the equation for \(K.E_2\):
\[ K.E_2 = \frac{1}{2} \cdot 0.004 \cdot \left(\frac{1}{4} v_1\right)^2 = \frac{1}{2} \cdot 0.004 \cdot \frac{1}{16} v_1^2 = \frac{0.004}{32} v_1^2 \]

7. **Calculate the ratio of kinetic energies**:
\[ \text{Ratio} = \frac{K.E_1}{K.E_2} = \frac{\frac{1}{2} \cdot 0.001 \cdot v_1^2}{\frac{0.004}{32} v_1^2} \]
- The \(v_1^2\) cancels out:
\[ = \frac{0.001/2}{0.004/32} = \frac{0.001 \cdot 32}{0.004 \cdot 2} = \frac{0.032}{0.008} = 4 \]

8. **Final Result**:
The ratio of their kinetic energies \(K.E_1 : K.E_2\) is:
\[ 4 : 1 \]