A super dense ball of very large mass is moving with a certain velocity in a particular direction . This heavy ball collides with a very light ball elastically . Calculate the approximate ratio of velocity of light object after collision to that with velocity of heavy ball .

Explanation

To solve the problem, we need to analyze the elastic collision between a heavy ball (mass \( M \)) and a light ball (mass \( m \)). We will use the principles of conservation of momentum and the properties of elastic collisions.

### Step-by-Step Solution:

1. **Define Initial Conditions:**
- Let the mass of the heavy ball be \( M \) and its initial velocity be \( u \).
- Let the mass of the light ball be \( m \) and its initial velocity be \( 0 \) (since it is at rest).

2. **Conservation of Momentum:**
The total momentum before the collision must equal the total momentum after the collision. Thus, we can write:
\[ Mu + m \cdot 0 = Mv_1 + mv_2 \]
Simplifying this gives us:
\[ Mu = Mv_1 + mv_2 \quad \text{(Equation 1)} \]

3. **Coefficient of Restitution:**
For an elastic collision, the coefficient of restitution \( e \) is equal to 1. This gives us the relationship between the velocities after and before the collision:
\[ \frac{v_2 - v_1}{u - 0} = 1 \]
Rearranging this, we find:
\[ v_2 - v_1 = u \quad \text{(Equation 2)} \]

4. **Solving the Equations:**
Now we have two equations:
- From Equation 1: \( Mu = Mv_1 + mv_2 \)
- From Equation 2: \( v_2 = v_1 + u \)

Substitute Equation 2 into Equation 1:
\[ Mu = Mv_1 + m(v_1 + u) \]
Expanding this gives:
\[ Mu = Mv_1 + mv_1 + mu \]
Rearranging terms:
\[ Mu - mu = (M + m)v_1 \]
Thus, we can solve for \( v_1 \):
\[ v_1 = \frac{Mu - mu}{M + m} \]

5. **Finding \( v_2 \):**
Substitute \( v_1 \) back into Equation 2 to find \( v_2 \):
\[ v_2 = v_1 + u = \frac{Mu - mu}{M + m} + u \]
Simplifying this gives:
\[ v_2 = \frac{Mu - mu + u(M + m)}{M + m} = \frac{Mu + uM}{M + m} = \frac{(M + m)u}{M + m} = 2u \]

6. **Calculating the Ratio:**
Now we have:
- \( v_1 = \frac{Mu - mu}{M + m} \)
- \( v_2 = 2u \)

We need the ratio of the velocities of the light ball to the heavy ball after the collision:
\[ \text{Ratio} = \frac{v_2}{v_1} = \frac{2u}{u} = 2 \]

### Final Answer:
The approximate ratio of the velocity of the light object after the collision to that of the heavy ball is \( 2:1 \).