One nitrogen molecule $P$, moving at a speed of $320\,\text{m s}^{-1}$ in a vacuum, collides with a nitrogen molecule $Q$ that is at rest. Following the collision, $P$ moves with a velocity of $180\,\text{m s}^{-1}$ at an angle of $55^{\circ}$ to its original direction. $Q$ moves in a direction making an angle of $34^{\circ}$ with the initial direction of $P$. Assume that no external forces act on the molecules. What is the magnitude $v$ of the velocity of $Q$ after the collision?
- A$120\,\text{m s}^{-1}$
- B$140\,\text{m s}^{-1}$
- C$180\,\text{m s}^{-1}$
- D$260\,\text{m s}^{-1}$