A sphere of mass `m_(1)` in motion hits directly another sphere of mass `m_(2)` at rest and sticks to it, the total kinetic energy after collision is `2//3` of their total K.E. before collision. Find the ratio of `m_(1) : m_(2)`.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the variables
Let:
- \( m_1 \) = mass of the moving sphere
- \( m_2 \) = mass of the sphere at rest
- \( u \) = initial velocity of the moving sphere
- \( v \) = final velocity of both spheres after collision (since they stick together)

### Step 2: Apply the conservation of momentum
According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.

Before the collision:
- Momentum = \( m_1 \cdot u + m_2 \cdot 0 = m_1 u \)

After the collision:
- Momentum = \( (m_1 + m_2) \cdot v \)

Setting these equal gives:
\[ m_1 u = (m_1 + m_2) v \]

From this, we can express \( v \):
\[ v = \frac{m_1}{m_1 + m_2} u \]

### Step 3: Calculate the initial and final kinetic energies
Initial kinetic energy (KE_initial):
\[ KE_{\text{initial}} = \frac{1}{2} m_1 u^2 \]

Final kinetic energy (KE_final):
\[ KE_{\text{final}} = \frac{1}{2} (m_1 + m_2) v^2 \]

Substituting \( v \) from Step 2 into the final kinetic energy equation:
\[ KE_{\text{final}} = \frac{1}{2} (m_1 + m_2) \left(\frac{m_1}{m_1 + m_2} u\right)^2 \]
\[ = \frac{1}{2} (m_1 + m_2) \frac{m_1^2}{(m_1 + m_2)^2} u^2 \]
\[ = \frac{1}{2} \cdot \frac{m_1^2}{m_1 + m_2} u^2 \]

### Step 4: Set up the equation using the given condition
We know from the problem that the total kinetic energy after the collision is \( \frac{2}{3} \) of the total kinetic energy before the collision:
\[ KE_{\text{final}} = \frac{2}{3} KE_{\text{initial}} \]

Substituting the expressions we found:
\[ \frac{1}{2} \cdot \frac{m_1^2}{m_1 + m_2} u^2 = \frac{2}{3} \cdot \frac{1}{2} m_1 u^2 \]

### Step 5: Simplify the equation
Cancelling \( \frac{1}{2} u^2 \) from both sides:
\[ \frac{m_1^2}{m_1 + m_2} = \frac{2}{3} m_1 \]

Multiplying both sides by \( (m_1 + m_2) \):
\[ m_1^2 = \frac{2}{3} m_1 (m_1 + m_2) \]

### Step 6: Rearranging the equation
Expanding and rearranging gives:
\[ 3m_1^2 = 2m_1^2 + 2m_2 m_1 \]
\[ 3m_1^2 - 2m_1^2 = 2m_2 m_1 \]
\[ m_1^2 = 2m_2 m_1 \]

Dividing both sides by \( m_1 \) (assuming \( m_1 \neq 0 \)):
\[ m_1 = 2m_2 \]

### Step 7: Find the ratio of \( m_1 \) to \( m_2 \)
Thus, the ratio of \( m_1 \) to \( m_2 \) is:
\[ \frac{m_1}{m_2} = 2 \quad \text{or} \quad m_1 : m_2 = 2 : 1 \]

### Final Answer
The ratio of \( m_1 : m_2 \) is \( 2 : 1 \).

---