Particles x (of mass 4 kg) and y (of mass 9 kg) move directly towards each otyher, collide and then separate. If `Deltav_(x)` is the change in the velocity of x and `Deltav_(y)` is the change in velocity of y then the magnitude of `(Deltav_(x))/(Deltav_(y))` is :

Options

A. `(9)/(4)`
B. `(3)/(2)`
C. `(2)/(3)`
D. `(4)/(9)`

Answer

To solve the problem, we will use the principle of conservation of momentum. Here are the steps to find the magnitude of the ratio of the changes in velocity of particles x and y after the collision.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let the mass of particle x, \( m_x = 4 \, \text{kg} \)
- Let the mass of particle y, \( m_y = 9 \, \text{kg} \)
- Let the initial velocity of particle x be \( v_{x1} \)
- Let the final velocity of particle x be \( v_{x2} \)
- Let the initial velocity of particle y be \( v_{y1} \)
- Let the final velocity of particle y be \( v_{y2} \)

2. **Write the Changes in Velocity**:
- The change in velocity for particle x is given by:
\[ \Delta v_x = v_{x2} - v_{x1} \]
- The change in velocity for particle y is given by:
\[ \Delta v_y = v_{y2} - v_{y1} \]

3. **Apply Conservation of Momentum**:
- According to the conservation of momentum:
\[ m_x v_{x1} + m_y v_{y1} = m_x v_{x2} + m_y v_{y2} \]
- Rearranging this gives:
\[ m_y v_{y2} - v_{y1} = m_x (v_{x2} - v_{x1}) \]

4. **Substitute the Changes in Velocity**:
- We can express the above equation in terms of changes in velocity:
\[ m_y \Delta v_y = -m_x \Delta v_x \]

5. **Express the Ratio of Changes in Velocity**:
- Rearranging the equation gives:
\[ \frac{\Delta v_x}{\Delta v_y} = -\frac{m_y}{m_x} \]
- Substituting the masses:
\[ \frac{\Delta v_x}{\Delta v_y} = -\frac{9}{4} \]

6. **Find the Magnitude**:
- The problem asks for the magnitude of the ratio:
\[ \left| \frac{\Delta v_x}{\Delta v_y} \right| = \frac{9}{4} \]

### Final Answer:
The magnitude of \( \frac{\Delta v_x}{\Delta v_y} \) is \( \frac{9}{4} \).

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