1. **State the problem:** We are given the equation $P_A = \left| \int v = m \rho V d\tau \right|$ and asked to solve for $v$ as a function of $t$. 2. **Clarify the equation:** The given expression is ambiguous. Assuming $v$ is velocity and $P_A$ relates to momentum or a similar quantity, and $m$, $\rho$, $V$, and $d\tau$ are constants or functions related to mass, density, volume, and differential volume element respectively. 3. **Assuming the integral represents total momentum:** $$P_A = \left| \int v \rho d\tau \right|$$ where $\rho$ is density and $v$ is velocity field over volume $V$. 4. **To find $v(t)$, we need a relation involving time:** If $v$ depends on $t$, and $P_A$ is known, then differentiating or expressing $v$ explicitly requires more information about $\rho$, $V$, and $P_A$ as functions of $t$. 5. **If $P_A$ is constant and $\rho$, $V$ are constants:** Then $$P_A = |v| \rho V$$ Solving for $v$: $$v = \frac{P_A}{\rho V}$$ 6. **If $P_A$ varies with time, then:** $$v(t) = \frac{P_A(t)}{\rho V}$$ 7. **Summary:** Without additional information about how $P_A$, $\rho$, and $V$ depend on time, the best we can do is express $v$ as: $$v(t) = \frac{P_A(t)}{\rho V}$$ This shows velocity as a function of time given the momentum $P_A(t)$, density $\rho$, and volume $V$. **Final answer:** $$v(t) = \frac{P_A(t)}{\rho V}$$