If one sphere collides head - on with another sphere of the same mass at rest inelastically. The ratio of their speeds `((v_(2))/(v_(1)))` after collision shall be

Options

A. `((1-e))/((1+e))`
B. `(2e)/((1+e))`
C. `((1+e))/((1-e))`
D. e

Answer

To solve the problem of determining the ratio of the speeds \( \frac{v_2}{v_1} \) after an inelastic collision between two spheres of equal mass, we can follow these steps:

### Step 1: Understand the scenario
We have two spheres, both with the same mass \( m \). One sphere is moving with an initial velocity \( u_1 \), and the other sphere is at rest (initial velocity \( u_2 = 0 \)). After the collision, both spheres move together with velocities \( v_1 \) and \( v_2 \).

### Step 2: Apply the conservation of momentum
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. Therefore, we can write:

\[ m u_1 + m u_2 = m v_1 + m v_2 \]

Since \( u_2 = 0 \), this simplifies to:

\[ m u_1 = m v_1 + m v_2 \]

Dividing through by \( m \) (assuming \( m \neq 0 \)) gives us:

\[ u_1 = v_1 + v_2 \tag{1} \]

### Step 3: Use the coefficient of restitution
The coefficient of restitution \( e \) for an inelastic collision is defined as:

\[ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} \]

In this case, since the collision is inelastic, \( e \) will be between 0 and 1. The relative velocity of approach is \( u_1 - 0 = u_1 \) and the relative velocity of separation is \( v_2 - v_1 \). Thus, we can write:

\[ e = \frac{v_2 - v_1}{u_1} \tag{2} \]

### Step 4: Substitute \( u_1 \) from equation (1) into equation (2)
From equation (1), we can express \( u_1 \) as \( v_1 + v_2 \). Substituting this into equation (2) gives:

\[ e = \frac{v_2 - v_1}{v_1 + v_2} \]

### Step 5: Rearranging the equation
We can rearrange the equation to isolate the ratio \( \frac{v_2}{v_1} \). Let \( x = \frac{v_2}{v_1} \). Then, we can express \( v_2 \) as \( x v_1 \). Substituting this into the equation gives:

\[ e = \frac{x v_1 - v_1}{v_1 + x v_1} = \frac{(x - 1)v_1}{(1 + x)v_1} \]

Cancelling \( v_1 \) (assuming \( v_1 \neq 0 \)) leads to:

\[ e = \frac{x - 1}{1 + x} \]

### Step 6: Solve for \( x \)
Cross-multiplying gives:

\[ e(1 + x) = x - 1 \]

Expanding and rearranging yields:

\[ ex + e = x - 1 \implies x - ex = e + 1 \implies x(1 - e) = 1 + e \]

Thus, we can solve for \( x \):

\[ x = \frac{1 + e}{1 - e} \]

### Step 7: Conclusion
The ratio of the speeds after the collision is:

\[ \frac{v_2}{v_1} = \frac{1 + e}{1 - e} \]