A ball of `4 kg` mass moving with a speed of `3ms^(-1)` has a head on elastic collision with a `6 kg` mass initially at rest. The speeds of both the bodies after collision are respectively

Options

A. `0.6 ms^(-1), 2.4 ms^(-1)`
B. `-0.6 ms^(-1), -2.4 ms^(-1)`
C. `-0.6 ms^(-1), 2.4 ms^(-1)`
D. `-0.6 ms^(-1), -2.4 ms^(-1)`

Explanation

To solve the problem of an elastic collision between a 4 kg ball moving at 3 m/s and a 6 kg ball initially at rest, we will use the principles of conservation of momentum and the properties of elastic collisions.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Mass of ball 1 (m1) = 4 kg
- Initial velocity of ball 1 (u1) = 3 m/s
- Mass of ball 2 (m2) = 6 kg
- Initial velocity of ball 2 (u2) = 0 m/s (since it is at rest)

2. **Conservation of Momentum:**
The total momentum before the collision is equal to the total momentum after the collision.
\[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \]
Substituting the known values:
\[ 4 \times 3 + 6 \times 0 = 4 v_1 + 6 v_2 \]
This simplifies to:
\[ 12 = 4 v_1 + 6 v_2 \quad \text{(Equation 1)} \]

3. **Elastic Collision Condition:**
For an elastic collision, the relative velocity of separation is equal to the relative velocity of approach.
\[ v_2 - v_1 = u_1 - u_2 \]
Substituting the known values:
\[ v_2 - v_1 = 3 - 0 \]
This simplifies to:
\[ v_2 - v_1 = 3 \quad \text{(Equation 2)} \]

4. **Solve Equations Simultaneously:**
From Equation 2, we can express \(v_2\) in terms of \(v_1\):
\[ v_2 = v_1 + 3 \]

Now substitute \(v_2\) in Equation 1:
\[ 12 = 4 v_1 + 6(v_1 + 3) \]
Expanding this gives:
\[ 12 = 4 v_1 + 6 v_1 + 18 \]
Combining like terms:
\[ 12 = 10 v_1 + 18 \]
Rearranging gives:
\[ 10 v_1 = 12 - 18 \]
\[ 10 v_1 = -6 \]
\[ v_1 = -0.6 \, \text{m/s} \]

5. **Find \(v_2\):**
Now substitute \(v_1\) back into the equation for \(v_2\):
\[ v_2 = -0.6 + 3 = 2.4 \, \text{m/s} \]

### Final Answer:
- The speed of the 4 kg ball after the collision (v1) is **-0.6 m/s** (indicating it moves in the opposite direction).
- The speed of the 6 kg ball after the collision (v2) is **2.4 m/s**.