A ball of mass 10 kg is moving with a velocity of 10 m / s . It strikes another ball of mass 5 kg which is moving in the same direction with a velocity of 4 m / s . If the collision is elastic, their velocities after the collision will be, respectively

Options

A. 6m/s, 12m/s
B. 12m/s, 6m/s
C. 12 m/s, 10 m/s
D. 12m/s, 25m/s

Answer

To solve the problem of two balls colliding elastically, we will use the principles of conservation of momentum and the coefficient of restitution. Here's a step-by-step solution:

### Step 1: Identify the given values
- Mass of ball 1 (m1) = 10 kg
- Initial velocity of ball 1 (u1) = 10 m/s
- Mass of ball 2 (m2) = 5 kg
- Initial velocity of ball 2 (u2) = 4 m/s

### Step 2: Write the conservation of momentum equation
The total momentum before the collision must equal the total momentum after the collision. This can be expressed as:
\[ m_1 \cdot u_1 + m_2 \cdot u_2 = m_1 \cdot v_1 + m_2 \cdot v_2 \]
Where:
- \( v_1 \) = final velocity of ball 1
- \( v_2 \) = final velocity of ball 2

Substituting the known values:
\[ 10 \cdot 10 + 5 \cdot 4 = 10 \cdot v_1 + 5 \cdot v_2 \]
\[ 100 + 20 = 10v_1 + 5v_2 \]
\[ 120 = 10v_1 + 5v_2 \] (Equation 1)

### Step 3: Write the coefficient of restitution equation
For an elastic collision, the coefficient of restitution (e) is defined as:
\[ e = \frac{\text{relative velocity after collision}}{\text{relative velocity before collision}} \]
Since the collision is elastic, \( e = 1 \).

The relative velocity before the collision is:
\[ u_1 - u_2 = 10 - 4 = 6 \, \text{m/s} \]

The relative velocity after the collision is:
\[ v_2 - v_1 \]

Setting up the equation:
\[ 1 = \frac{v_2 - v_1}{6} \]
This gives us:
\[ v_2 - v_1 = 6 \] (Equation 2)

### Step 4: Solve the system of equations
Now we have two equations:
1. \( 10v_1 + 5v_2 = 120 \) (Equation 1)
2. \( v_2 - v_1 = 6 \) (Equation 2)

From Equation 2, we can express \( v_2 \) in terms of \( v_1 \):
\[ v_2 = v_1 + 6 \]

Substituting this into Equation 1:
\[ 10v_1 + 5(v_1 + 6) = 120 \]
\[ 10v_1 + 5v_1 + 30 = 120 \]
\[ 15v_1 + 30 = 120 \]
\[ 15v_1 = 90 \]
\[ v_1 = 6 \, \text{m/s} \]

Now substituting \( v_1 \) back into Equation 2 to find \( v_2 \):
\[ v_2 = 6 + 6 = 12 \, \text{m/s} \]

### Final Result
The velocities after the collision are:
- \( v_1 = 6 \, \text{m/s} \) (for the 10 kg ball)
- \( v_2 = 12 \, \text{m/s} \) (for the 5 kg ball)