The given velocity function is $v(t) = 3 - \frac{10}{t+1}$, and we are tasked with finding $v(4)$.

To solve for $v(4)$, we substitute $t = 4$ into the function:

$v(4) = 3 - \frac{10}{4+1} = 3 - \frac{10}{5} = 3 - 2 = 1$

Thus, the value of $v(4)$ is $1$.

The correct answer is $\boxed{1}$.

Would you like more details on how the function was evaluated? Here are five follow-up questions:

1. How does the term $\frac{10}{t+1}$ affect the velocity function for different values of $t$?
2. What is the behavior of the velocity function as $t \to \infty$?
3. How would you find the time when the particle comes to rest?
4. Can you describe how the velocity changes as $t$ increases?
5. How could you graph the function $v(t) = 3 - \frac{10}{t+1}$?

Tip: Always substitute carefully, especially when dealing with fractions, to avoid simple arithmetic errors.