Astro Disastro, Cosmic Defender of Justice, has been captured on the planet Azzak by the evil Dr. Evad Nevals. Before Nevals kills Astro, however, the two share a drink while Nevals describes his plans for galactic domination. (Divulging your full plans to the one man who can stop you, just before killing him, is of course one of the oldest required rituals of the Brotherhood of Mad Geniuses. Everyone knows this.) The drinks are finished now, and Nevals has just told Astro that his (Astro's) drink was poisoned, and he has only thirteen hours to consume the antidote, which can only be found on the planet Bliffid.

Astro stages a daring escape and takes off in his Porsche 999 Star Cruiser at 95% of the speed of light toward Bliffid, which is 40,000,000,000 km away from Azzak.

Well, let's do the math now. To figure out how long it takes Astro to reach Bliffid, we simply divide the distance by the speed. (For example, if you drive 120 miles at 60 miles per hour, it takes 2 hours. 120/60 = 2.) The distance is 40,000,000,000 km, and Astro's speed is about 1,025,000,000 kph. Dividing these, we get about 39 hours. Astro only has thirteen hours, so things don't look good.

But wait! This is all from the point of view of a person standing on Azzak. Since the poison is metabolizing in Astro's body, we need to look at the situation from Astro's frame of reference. There are two ways we can do this, and they'll both lead to the same result.

1. Imagine a giant ruler stretching from Azzak to Bliffid. This ruler is 40,000,000,000 km long. However, from Astro's point of view this ruler is flying past Astro at nearly the speed of light. And we've learned that such an object is subject to length contraction. In Astro's frame of reference, the distance from Azzak to Bliffid is shrunk by a factor of gamma. At 95% of the speed of light (.95c), gamma is about 3.2, and so as far as Astro is concerned the distance is only about 12,500,000,000 km (40 billion divided by 3.2) away, and when we divide this by Astro's speed we get 12.2 hours. Astro reaches Bliffid with nearly an hour to spare!
   
   
2. The time it takes Astro to reach Bliffid, from the point of view of an observer on Azzak is about 39 hours, as we've already observed. However, this is a dilated time. We know that Astro's "clock" runs slower, by a factor of gamma (about 3.2, remember). So to calculate the time in Astro's frame of reference, we take the 39 hours and divide by 3.2. Again, this gives us 12.2 hours, leaving Astro nearly an hour. (Which is good, since it took 20 minutes for Astro to make his escape, and it will take about another 20 to find the antidote once he reaches Bliffid.)

As it turns out, there are a lot of subtleties involved in the situation I've just described, if one starts examining it in more detail. When you start thinking about it closely, eventually you arrive at a question which typically starts out something like: "Wait a minute. In Astro's frame of reference, Nevals's clock runs slower, so the trip should take longer in Astro's frame, not shorter...." If this question interests or bothers you, check out the page on "Temporal subtleties of Astro's adventure." Or you can just take my word for it that if you reason everything out correctly, everything really does work out okay, and skip to the "Mass and energy" page indicated below.