Three balls of masses `m_(1),m_(2),m_(3)` are lying in a straight line. The first ball is moved with a certain velocity so that it strikes the second ball directly, then the second ball collides with the third. If the coefficient of elasticity for each ball is e and after impact first ball comes to rest, while after second impact the second ball comes to rest. Then `m_(1),m_(2),m_(3)` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. None of these

Answer

To solve the problem, we need to analyze the collisions between the three balls step by step, applying the principles of conservation of momentum and the coefficient of restitution.

### Step 1: Analyze the first collision (between m1 and m2)

Let:
- \( m_1 \) be the mass of the first ball.
- \( m_2 \) be the mass of the second ball.
- \( u_1 \) be the initial velocity of \( m_1 \) (before the collision).
- \( u_2 = 0 \) be the initial velocity of \( m_2 \) (before the collision).
- \( v_1 \) be the final velocity of \( m_1 \) (after the collision).
- \( v_2 \) be the final velocity of \( m_2 \) (after the collision).

According to the problem, after the first collision, the first ball comes to rest, so:
\[ v_1 = 0 \]

Using the conservation of momentum for the first collision:
\[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \]
Substituting the known values:
\[ m_1 u_1 + 0 = 0 + m_2 v_2 \]
This simplifies to:
\[ m_1 u_1 = m_2 v_2 \quad (1) \]

Next, we apply the coefficient of restitution (e) for the first collision:
\[ e = \frac{v_2 - v_1}{u_1 - u_2} \]
Substituting the known values:
\[ e = \frac{v_2 - 0}{u_1 - 0} \]
This gives us:
\[ v_2 = e u_1 \quad (2) \]

### Step 2: Substitute equation (2) into equation (1)

From equation (1):
\[ m_1 u_1 = m_2 (e u_1) \]
We can cancel \( u_1 \) (assuming \( u_1 \neq 0 \)):
\[ m_1 = m_2 e \quad (3) \]

### Step 3: Analyze the second collision (between m2 and m3)

Let:
- \( m_3 \) be the mass of the third ball.
- \( u_3 = 0 \) be the initial velocity of \( m_3 \).
- \( v_3 \) be the final velocity of \( m_3 \) (after the collision).

After the second collision, the second ball comes to rest, so:
\[ v_2 = 0 \]

Using the conservation of momentum for the second collision:
\[ m_2 v_2 + m_3 u_3 = m_2 v_2 + m_3 v_3 \]
Substituting the known values:
\[ m_2 (e u_1) + 0 = 0 + m_3 v_3 \]
This simplifies to:
\[ m_2 e u_1 = m_3 v_3 \quad (4) \]

Again, applying the coefficient of restitution for the second collision:
\[ e = \frac{v_3 - v_2}{u_2 - u_3} \]
Substituting the known values:
\[ e = \frac{v_3 - 0}{(e u_1) - 0} \]
This gives us:
\[ v_3 = e (e u_1) = e^2 u_1 \quad (5) \]

### Step 4: Substitute equation (5) into equation (4)

From equation (4):
\[ m_2 e u_1 = m_3 (e^2 u_1) \]
We can cancel \( u_1 \) (assuming \( u_1 \neq 0 \)):
\[ m_2 e = m_3 e^2 \]
Dividing both sides by \( e \) (assuming \( e \neq 0 \)):
\[ m_2 = m_3 e \quad (6) \]

### Step 5: Relate all masses together

From equations (3) and (6):
1. \( m_1 = m_2 e \)
2. \( m_2 = m_3 e \)

Substituting equation (6) into equation (3):
\[ m_1 = (m_3 e) e = m_3 e^2 \]

Thus, we have:
- \( m_1 = m_3 e^2 \)
- \( m_2 = m_3 e \)
- \( m_3 = m_3 \)

### Conclusion

The relationship between the masses can be summarized as:
\[ m_1 : m_2 : m_3 = e^2 : e : 1 \]