Two substances have masses 90kg and 160kg respectively. They possess equal kinetic energy. The ratio of their momenta will be

Options

A. `(3)/(4)`
B. `(9)/(16)`
C. `(9)/(4)`
D. `(4)/(9)`

Explanation

To solve the problem of finding the ratio of the momenta of two substances with equal kinetic energy, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Mass of substance 1, \( m_1 = 90 \, \text{kg} \)
- Mass of substance 2, \( m_2 = 160 \, \text{kg} \)
- Both substances have equal kinetic energy.

2. **Write the Formula for Kinetic Energy:**
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
For substance 1:
\[ KE_1 = \frac{1}{2} m_1 v_1^2 \]
For substance 2:
\[ KE_2 = \frac{1}{2} m_2 v_2^2 \]

3. **Set the Kinetic Energies Equal:**
Since both substances have equal kinetic energy:
\[ \frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_2 v_2^2 \]
The \( \frac{1}{2} \) cancels out:
\[ m_1 v_1^2 = m_2 v_2^2 \]

4. **Rearrange to Find the Velocity Ratio:**
Rearranging gives:
\[ \frac{v_1^2}{v_2^2} = \frac{m_2}{m_1} \]
Taking the square root of both sides:
\[ \frac{v_1}{v_2} = \sqrt{\frac{m_2}{m_1}} \]

5. **Substituting the Mass Values:**
Substitute \( m_1 = 90 \, \text{kg} \) and \( m_2 = 160 \, \text{kg} \):
\[ \frac{v_1}{v_2} = \sqrt{\frac{160}{90}} = \sqrt{\frac{16}{9}} = \frac{4}{3} \]

6. **Calculate the Momentum:**
The momentum \( p \) of an object is given by:
\[ p = mv \]
Thus, the momenta of the two substances are:
\[ p_1 = m_1 v_1 \quad \text{and} \quad p_2 = m_2 v_2 \]

7. **Find the Ratio of the Momentums:**
The ratio of the momenta is:
\[ \frac{p_1}{p_2} = \frac{m_1 v_1}{m_2 v_2} \]
Substituting \( v_1 = \frac{4}{3} v_2 \):
\[ \frac{p_1}{p_2} = \frac{m_1 \left(\frac{4}{3} v_2\right)}{m_2 v_2} = \frac{m_1 \cdot \frac{4}{3}}{m_2} \]
Simplifying gives:
\[ \frac{p_1}{p_2} = \frac{90 \cdot \frac{4}{3}}{160} = \frac{120}{160} = \frac{3}{4} \]

### Final Answer:
The ratio of their momenta is:
\[ \frac{p_1}{p_2} = \frac{3}{4} \]