A sphere of mass m moving with a constant velocity hits another stationary sphere of the same mass, if e is the coefficient of restitution, then ratio of speed of the first sphere to the speed of the second sphere after collision will be

Options

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

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have two spheres of equal mass \( m \). One sphere is moving with a constant velocity \( v \) and the other is stationary. After the collision, we need to find the ratio of the speed of the first sphere \( v_1 \) to the speed of the second sphere \( v_2 \) given the coefficient of restitution \( e \).

### Step 2: Apply the conservation of momentum
The total momentum before the collision must equal the total momentum after the collision.

- Initial momentum:
\[ p_{\text{initial}} = mv + 0 = mv \]

- Final momentum:
\[ p_{\text{final}} = mv_1 + mv_2 \]

Setting these equal gives us:
\[ mv = mv_1 + mv_2 \]
Dividing through by \( m \) (since \( m \neq 0 \)):
\[ v = v_1 + v_2 \quad \text{(Equation 1)} \]

### Step 3: Apply the coefficient of restitution
The coefficient of restitution \( e \) is defined as the ratio of the relative speed of separation to the relative speed of approach.

- Velocity of separation:
\[ v_{\text{separation}} = v_2 - v_1 \]

- Velocity of approach:
\[ v_{\text{approach}} = v - 0 = v \]

Thus, we have:
\[ e = \frac{v_2 - v_1}{v} \]
Rearranging gives us:
\[ v_2 - v_1 = ev \quad \text{(Equation 2)} \]

### Step 4: Solve the equations simultaneously
Now we have two equations:
1. \( v_1 + v_2 = v \)
2. \( v_2 - v_1 = ev \)

We can add these two equations:
\[ (v_1 + v_2) + (v_2 - v_1) = v + ev \]
This simplifies to:
\[ 2v_2 = v(1 + e) \]
Thus, we find:
\[ v_2 = \frac{(1 + e)v}{2} \]

Now, substituting \( v_2 \) back into Equation 1 to find \( v_1 \):
\[ v_1 + \frac{(1 + e)v}{2} = v \]
Rearranging gives:
\[ v_1 = v - \frac{(1 + e)v}{2} = \frac{2v - (1 + e)v}{2} = \frac{(1 - e)v}{2} \]

### Step 5: Find the ratio \( \frac{v_1}{v_2} \)
Now we can find the ratio of the speeds:
\[ \frac{v_1}{v_2} = \frac{\frac{(1 - e)v}{2}}{\frac{(1 + e)v}{2}} = \frac{1 - e}{1 + e} \]

### Final Answer
The ratio of the speed of the first sphere to the speed of the second sphere after the collision is:
\[ \frac{v_1}{v_2} = \frac{1 - e}{1 + e} \]