Point $P_1$ is located $600$ miles West of point $P_2$. At $7:00\text{AM}$ a car departs from $P_1$ and drives East at a speed of $50$mph. At $8:00\text{AM}$ another car departs from $P_2$ and drives West at a constant speed of $x$ miles per hour. If the cars meet each other exactly halfway between $P_1$ and $P_2$, what is the value of $x$?
1. **Determine the distance to the midpoint:**
   The total distance between $P_1$ and $P_2$ is $600$ miles. The midpoint is therefore $300$ miles from each point.

2. **Calculate the time taken by the first car to reach the midpoint:**
   The first car departs from $P_1$ at $7:00\text{AM}$ and drives East at a speed of $50$ mph. The distance to the midpoint is $300$ miles.
   \[
   \text{Time taken by the first car} = \frac{\text{Distance}}{\text{Speed}} = \frac{300 \text{ miles}}{50 \text{ mph}} = 6 \text{ hours}
   \]
   Therefore, the first car reaches the midpoint at:
   \[
   7:00\text{AM} + 6 \text{ hours} = 1:00\text{PM}
   \]

3. **Determine the time available for the second car to reach the midpoint:**
   The second car departs from $P_2$ at $8:00\text{AM}$ and needs to reach the midpoint by $1:00\text{PM}$. The time available for the second car is:
   \[
   1:00\text{PM} - 8:00\text{AM} = 5 \text{ hours}
   \]

4. **Calculate the required speed of the second car:**
   The second car needs to cover the same distance of $300$ miles in $5$ hours.
   \[
   \text{Speed of the second car} = \frac{\text{Distance}}{\text{Time}} = \frac{300 \text{ miles}}{5 \text{ hours}} = 60 \text{ mph}
   \]

Conclusion:
The value of $x$ is $60$ mph.

The final answer is $\boxed{60}$.