A sphere of mass m moving with a uniform velocity v hits another identical sphere at rest. If the coefficient of restitution is e, the ratio of the velocities of the two sphere after the collision is `(e-1)/(e+1)`.

Answer

To solve the problem step by step, we will analyze the collision of two identical spheres, one moving and the other at rest, and use the principles of conservation of momentum and the coefficient of restitution.

### Step 1: Define the Variables
Let:
- Mass of sphere 1 (m1) = m
- Mass of sphere 2 (m2) = m
- Initial velocity of sphere 1 (u1) = v (moving with uniform velocity)
- Initial velocity of sphere 2 (u2) = 0 (at rest)
- Final velocity of sphere 1 after collision (v1)
- Final velocity of sphere 2 after collision (v2)
- Coefficient of restitution = e

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

\[ m \cdot v + m \cdot 0 = m \cdot v_1 + m \cdot v_2 \]

This simplifies to:

\[ v = v_1 + v_2 \]

(Equation 1)

### Step 3: Apply the Coefficient of Restitution
The coefficient of restitution (e) is defined as the ratio of the relative velocity of separation to the relative velocity of approach. Thus, we can write:

\[ e = \frac{v_2 - v_1}{u_1 - u_2} \]

Substituting the known values:

\[ e = \frac{v_2 - v_1}{v - 0} \]

This can be rearranged to:

\[ v_2 - v_1 = e \cdot v \]

(Equation 2)

### Step 4: Solve the Equations
Now we have two equations:
1. \( v = v_1 + v_2 \) (Equation 1)
2. \( v_2 - v_1 = e \cdot v \) (Equation 2)

From Equation 1, we can express \( v_2 \) in terms of \( v_1 \):

\[ v_2 = v - v_1 \]

Substituting this into Equation 2:

\[ (v - v_1) - v_1 = e \cdot v \]

This simplifies to:

\[ v - 2v_1 = e \cdot v \]

Rearranging gives:

\[ 2v_1 = v - e \cdot v \]

\[ 2v_1 = v(1 - e) \]

Thus,

\[ v_1 = \frac{v(1 - e)}{2} \]

### Step 5: Find \( v_2 \)
Now substituting \( v_1 \) back into Equation 1 to find \( v_2 \):

\[ v_2 = v - v_1 = v - \frac{v(1 - e)}{2} \]

This simplifies to:

\[ v_2 = v \left(1 - \frac{1 - e}{2}\right) = v \left(\frac{2 - (1 - e)}{2}\right) = v \left(\frac{1 + e}{2}\right) \]

### Step 6: Calculate the Ratio of Velocities
Now we have:

- \( v_1 = \frac{v(1 - e)}{2} \)
- \( v_2 = \frac{v(1 + e)}{2} \)

The ratio of the velocities \( \frac{v_1}{v_2} \) is:

\[ \frac{v_1}{v_2} = \frac{\frac{v(1 - e)}{2}}{\frac{v(1 + e)}{2}} = \frac{1 - e}{1 + e} \]

### Conclusion
The statement given in the question is that the ratio of the velocities of the two spheres after the collision is \( \frac{e - 1}{e + 1} \). Our derived ratio is \( \frac{1 - e}{1 + e} \), which shows that the statement is false.

### Final Answer
The answer is **False**.