A block of mass m moving with a certain speed collides with another identical block at rest . If coefficient of restitution is `1/(sqrt(2))` then find ratio of intial kinetic energy to kinetic energy lost in collision .

Explanation

To solve the problem, we need to find the ratio of initial kinetic energy to the kinetic energy lost during the collision of two identical blocks, given that the coefficient of restitution is \( \frac{1}{\sqrt{2}} \).

### Step-by-step Solution:

1. **Define the Initial Conditions:**
- Let the mass of each block be \( m \).
- The initial speed of the moving block is \( u \).
- The second block is at rest, so its initial speed is \( 0 \).

2. **Calculate Initial Kinetic Energy:**
- The initial kinetic energy \( KE_i \) of the system is given by:
\[ KE_i = \frac{1}{2} m u^2 \]

3. **Conservation of Momentum:**
- According to the conservation of momentum, the total momentum before the collision equals the total momentum after the collision:
\[ mu + 0 = mv_1 + mv_2 \]
- This simplifies to:
\[ v_1 + v_2 = u \quad \text{(Equation 1)} \]

4. **Coefficient of Restitution:**
- The coefficient of restitution \( e \) is defined as:
\[ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} \]
- For our case:
\[ e = \frac{v_2 - v_1}{u - 0} = \frac{v_2 - v_1}{u} \]
- Given \( e = \frac{1}{\sqrt{2}} \), we can write:
\[ v_2 - v_1 = \frac{u}{\sqrt{2}} \quad \text{(Equation 2)} \]

5. **Solving the Equations:**
- From Equation 1, we have \( v_2 = u - v_1 \).
- Substitute \( v_2 \) in Equation 2:
\[ (u - v_1) - v_1 = \frac{u}{\sqrt{2}} \]
\[ u - 2v_1 = \frac{u}{\sqrt{2}} \]
- Rearranging gives:
\[ 2v_1 = u - \frac{u}{\sqrt{2}} \]
\[ v_1 = \frac{u}{2} \left(1 - \frac{1}{\sqrt{2}}\right) \]

6. **Finding \( v_2 \):**
- Substitute \( v_1 \) back into Equation 1 to find \( v_2 \):
\[ v_2 = u - v_1 = u - \frac{u}{2} \left(1 - \frac{1}{\sqrt{2}}\right) \]

7. **Calculating Final Kinetic Energy:**
- The final kinetic energy \( KE_f \) is:
\[ KE_f = \frac{1}{2} m v_1^2 + \frac{1}{2} m v_2^2 \]

8. **Kinetic Energy Lost:**
- The kinetic energy lost \( \Delta KE \) is:
\[ \Delta KE = KE_i - KE_f \]

9. **Finding the Ratio:**
- The ratio of initial kinetic energy to kinetic energy lost is:
\[ \text{Ratio} = \frac{KE_i}{\Delta KE} \]

10. **Final Calculation:**
- Substitute the values to find the ratio.

### Final Answer:
The ratio of initial kinetic energy to kinetic energy lost in the collision is \( 4:1 \).