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Old June 14 2013, 10:11 AM   #4221
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BillJ wrote: View Post
As far as plot-holes go...

When the Enterprise starts what I've seen called her improbable fall to Earth, couldn't she have actually been in the planet's orbital path? At roughly 67,000 miles per hour and factoring in the time it took for New Vulcan to be called, the Vengeance repaired, torpedoes to be sabotaged I don't think the fall is as unlikely as when I saw the film before.

And then! To top it off, they start crashing into, what? Not the Moon, the Earth! I *think* some character mumbles something about them being caught in the Earth’s gravity, and they are all the sudden being pulled in. Here’s where the movie gets a little cloudy, or maybe it is just my understanding of what was supposed to be going on. Within a few minutes they are pulled from right next to the Moon all the way into the Earth’s atmosphere. This is just insane.

Rather than go through some equations about how long this would take, instead we can just look at the case of Apollo 13. We launched a rocket from Earth, and due to a catastrophic failure of one of the oxygen tanks, they had to abort their lunar landing mission and move to a “free return” trajectory around the moon, and back to Earth. “Free return” just means that you don’t have to fire the rockets to return to Earth, you just use lunar gravity to swing you back around to Earth, with some minor course corrections. This is cool, because it tells us how long it takes a spaceship to “fall” back to Earth from the moon, if it can’t use its engines! In the case of Apollo 13, it took about 64 hours. Actually, that’s faster than it otherwise would have, because they did burn the descent engine two hours after swinging around the moon to speed their return to Earth by 10 hours. Anyway, the bottom line is that it takes *days*, not minutes, for a spacecraft to fall to Earth from lunar orbit.

When the Enterprise is falling into the Earth, it looks like it is falling straight into it, as if the two are balls on a string being drawn to each other. That’s not the way two bodies gravitationally attracted to each other work -- they approach each other on curved paths. Have you ever wondered why everything in space orbits something? It is because of conservation of angular momentum.

The most famous terrestrial example is an ice skater spinning. When she draws in her outstretched arms, she starts spinning faster and faster. The same thing would happen to the Enterprise as it fell to Earth. It wouldn’t fall straight in, it would kind of orbit. If it didn’t have the energy to make a complete orbit, it would still sort of half-loop around the Earth, and come in at an angle.

What energy would it have? Ignoring vectors, the formula for angular momentum (L) is L=rmv, where r is the distance from the axis of rotation to the thing rotating around it, m is the mass, and v is the velocity perpendicular to the line defined by r. We can make a ratio of the angular momentum at the Moon’s orbit and the angular momentum as the Enterprise enters the Earth’s atmosphere. Then, since angular momentum is conserved, and mass is conserved, these two quantities cancel. We’re left with vE=vM*(rM/rE). The ratio of the distance to the Moon to the Earth’s radius is about 60. That means whatever transverse velocity the Enterprise had at the orbit of the Moon would be amplified by a factor of 60 by the time it reached Earth, just due to conservation of angular momentum.

Would they have even crashed into the Earth? The escape velocity of the Earth (the velocity needed to achieve orbit) is about 7 km/s if you are already in space (it is higher if you have to leave the surface). And 7 km/s / 60 = about 100 m/s. So if the Enterprise was traveling at 100 meters per second or more relative to the Earth when it was at the Moon’s orbit, it never would have fallen all the way to the Earth, it would have attained orbital velocity by the time it reached the atmosphere. One hundred meters per second is not very fast -- that’s only ten times faster than a human can run! That’s nothing for a ship that just dropped out of warp and is being hit by projectiles. Just shoot a photon torpedo in the opposite direction and let the back reaction give you the tiny push to remain in orbit.

Fine, so you have to have them actually crash into Earth, because the script calls for it. My point is, show them streaking into the atmosphere at an angle, not falling directly down on the Earth. It is a small thing, but to anyone who knows science, it is glaring, and just shows that most people who worked on this movie know very little about physics and didn’t talk to anyone who did.
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