Two balls of equal mass, 0.5 kg, collide. The first ball has an initial velocity of 10 m/s, and the second is at rest. If they move off together, what is their final velocity?

We are given:

• Mass of the first ball, $m_1 = 0.5 \text{ kg}$
• Initial velocity of the first ball, $u_1 = 10 \text{ m/s}$
• Mass of the second ball, $m_2 = 0.5 \text{ kg}$ (since it has equal mass)
• Initial velocity of the second ball, $u_2 = 0 \text{ m/s}$ (at rest)
• The balls move off together, indicating a perfectly inelastic collision. We need to find their common final velocity, $v$.

For a perfectly inelastic collision, the total momentum before the collision is equal to the total momentum after the collision.
The formula is:

$$m_1 u_1 + m_2 u_2 = (m_1 + m_2) v$$

$$5 \text{ kg} \cdot \text{m/s} + 0 \text{ kg} \cdot \text{m/s} = (1.0 \text{ kg}) v$$ $$5 \text{ kg} \cdot \text{m/s} = (1.0 \text{ kg}) v$$

$$v = \frac{5 \text{ kg} \cdot \text{m/s}}{1.0 \text{ kg}}$$ $$v = 5 \text{ m/s}$$

The final velocity with which they move off together is $5 \text{ m/s}$.

Comparing this to the given options:
A. $20 \text{ m/s}$
B. $10 \text{ m/s}$
C. $5 \text{ m/s}$
D. $0.5 \text{ m/s}$

The correct option is C.

The final answer is $\boxed{\text{C. } 5 \text{ m/s}}$.