Final answer:
To determine whether the USS Enterprise will reach the space station after engine failure, we must consider its momentum and potential backup systems. Lack of details about its condition makes a definitive answer impossible, but if inertia and its trajectory are favorable, it could coast to its destination.
Step-by-step explanation:
To determine if the USS Enterprise will make it to the repair facility space station after the engine stops 500 km out, we need to consider the principles of physics that apply to space travel.
Once a spacecraft has escaped Earth's gravity and is en route to its destination, it follows an orbit around the Sun. If no other forces act upon it, the ship will continue to coast along this trajectory due to Newton's first law of motion, which states that an object in motion stays in motion unless acted upon by an external force (inertia).
Since the question doesn't provide details about whether the USS Enterprise has momentum or whether it can still use thrusters or alternative methods to propel itself, we cannot give a definitive answer.
However, if the ship had enough initial velocity directed toward the space station and there are no significant other forces (like gravitational pull from other bodies), it could potentially coast the remaining 500 km to the station. Additionally, since space crafts are designed with a high degree of autonomy and reliability, they may have backup systems that could potentially be used in situations like this.