Masses of two substances are 1 g and 9 g respectively. If their kinetic energies are same, then the ratio of their momentum will be

Options

A. `1:9`
B. `9:1`
C. `3:1`
D. `1:3`

Explanation

To solve the problem, we need to find the ratio of the momentum of two substances given that their masses are 1 g and 9 g, and their kinetic energies are the same.

### Step-by-Step Solution:

1. **Define the variables:**
- Let the mass of the first substance (m1) = 1 g
- Let the mass of the second substance (m2) = 9 g
- Let the kinetic energy of both substances be the same (K).

2. **Write the formula for kinetic energy:**
The kinetic energy (K) of an object is given by the formula:
\[ K = \frac{1}{2} mv^2 \]
where \( m \) is the mass and \( v \) is the velocity.

3. **Set up the equations for kinetic energy:**
For the first substance:
\[ K = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} (1) v_1^2 = \frac{1}{2} v_1^2 \]
For the second substance:
\[ K = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} (9) v_2^2 = \frac{9}{2} v_2^2 \]

4. **Equate the kinetic energies:**
Since both kinetic energies are equal:
\[ \frac{1}{2} v_1^2 = \frac{9}{2} v_2^2 \]

5. **Simplify the equation:**
Multiply both sides by 2 to eliminate the fraction:
\[ v_1^2 = 9 v_2^2 \]

6. **Take the square root to find the ratio of velocities:**
\[ v_1 = 3 v_2 \]

7. **Calculate the momentum:**
The momentum (p) is given by:
\[ p = mv \]
For the first substance:
\[ p_1 = m_1 v_1 = 1 \cdot v_1 = v_1 \]
For the second substance:
\[ p_2 = m_2 v_2 = 9 \cdot v_2 \]

8. **Substitute the value of \( v_1 \):**
Substitute \( v_1 = 3 v_2 \) into the equation for \( p_1 \):
\[ p_1 = 3 v_2 \]

9. **Find the ratio of momenta:**
Now, we can find the ratio of the momenta:
\[ \frac{p_1}{p_2} = \frac{3 v_2}{9 v_2} = \frac{3}{9} = \frac{1}{3} \]

10. **Conclusion:**
The ratio of the momentum of the two substances is:
\[ p_1 : p_2 = 1 : 3 \]