In Alien Resurrection, the science ship was parked in "unregulated space" while they did experiments. Then, when the aliens get loose, the ship starts an autopilot trip program to fly back to Earth. When Ripley asks how long until they get to Earth, a soldier says 3 hours. The next shot is of the ship passing by Jupiter on its way. The ship does not look to be traveling at the speed of light! So how fast would the ship need to travel to go from, we'll say, slightly beyond Jupiter to Earth in only 3 hours?
Jupiter averages about 43 light-minutes from the Sun, Earth is about 8.3 light-minutes. That means, at most, it's 51.3 light-minutes from Earth to Jupiter, and at a minimum 34.7 light-minutes.
So, simple math. The ship would be traveling at between 0.193 and 0.285 c to take three hours to cover the distance, depending on the distance between Earth and Jupiter at the time.
Depends where Earth and Jupiter are in their orbits.
If Jupiter is in opposition, it is abt 390 million miles from Earth, so the ship would need to travel at 130 million mph - just under 0.2c.
If they are on opposite sides of the Sun (superior conjunction) they are a bit over 570 million miles apart, so a speed of 190 million mph would be required - just over 0.28c.
Here is a nice app here for calculating exactly this kind of thing: http://convertalot.com/relativistic_star_ship_calculator.html
First the space ship must accelerate with full power, and when it has reached half-way, it has to decelerate with full power in order to stop at earth.
What the other comments fail to take into account is relativity. Time will move slower on the spaceship than on Earth, so if it takes 3 hours on the spaceship, the time passed on earth would have been ~3 hours 10 minutes.
The acceleration would also have to be insanely high, at ~3150 g (1 g ~=9.82 m/s)
Traveling the 6.168 AU (if opposite sides of the Sun, otherwise use 4.172 AU if they are closest to each other), the following values apply:
Acceleration: 3150 g
Time on spaceship: 3 hours
Time on Earth: 3 hours 10 min
Maximum speed relative to Earth: 0.5 c (1 c = 299792458 m/s, speed of light)
Rule of thumb gives that when speed is above ~0.1 c, you need to take relativity into account.
The above was under assumption that the spaceship will stop at Earth, but in the movie the spaceship was supposed to crash. In that case, just increase the distance, 6.168 AU, by a factor two, and adjust the acceleration such that the time on the ship becomes 6 hours. Earth would then be reached after 3 hours on the ship, and the maximum speed indicated is the speed it would impact Earth with.
As has been answered by others, the average speed would have to be at least 0.2c (or 60,000 km/s) over a distance of 624,000,000 km (in round numbers). With even acceleration all the way, the end speed would have to be 0.4c, which means that the acceleration would have to be 1,154 gees. That's a pretty hefty acceleration. It gets worse if the ship accelerates to the midpoint and then decelerates. Then the speed of 0.4c would have to be reached at the midpoint, which means double the acceleration.