“Not yet.” Rick’s skepticism increased. To tell the truth, he had mentally given up on the problem. It didn’t just call for straight learning of facts. You had to understand loads of math and orbital equations, which even Gina Styan admitted was a job for specialists. He wasn’t sure where to start. “Do you know how to do it?”
It was a question designed to get rid of her. He was convinced that Alice would have no idea. She would admit her ignorance, and then he’d ask her to go away. But she was pursing up her mouth and wrinkling her nose.
“I don’t know.” The grey, wide eyes focused on Rick—for the first time, she seemed to be looking at him and not through him. “Back when we were studying for our theory test, Turkey Gossage told me something that he said I ought to keep in mind for future reference. I have, but I don’t know what to do with it. That’s why I came to you. You’re smart, you can see through things.”
She was praising him, buttering him up. Rick knew that, but still he was pleased. Flattery, even when you recognized it, made you feel good. Maybe because it showed somebody thought you were worth flattering.
“What did Turkey tell you?”
“He said that the Sun’s gravity controls the movements of every body in the solar system.”
“Big deal. We all know that.”
“Yes. But he went on to say that the real surprise was how small the Sun’s gravitational force is. For instance, he told me that if you work out the acceleration on an object at Earth’s distance from the Sun, you find that it’s less than a thousandth of a gee. And when you get out to the Belt, it’s less than one ten-thousandth of a gee.”
“So?” But Rick already had the conviction that Alice was on to something.
“Well, we were told to assume that this ship is accelerating at a quarter of a gee.”
“I believe that, it certainly feels about right.”
“But that means once we get well away from Earth, the gravity forces from Earth and Sun and every other planet are nothing, compared with the ship’s own acceleration. I feel sure this is relevant to the problem—only I don’t see how to use it.”
“But I think I do! Give me a minute.” Rick sat staring at nothing. If all other accelerations on the ship were hundreds of times smaller than the ship’s own acceleration—
“Alice, we don’t need to know a whole slew of orbital theory. Our own ship’s acceleration is so high, we can get a good value for our travel time by assuming that we travel in a straight line and ignoring everything else. All we need is our own acceleration, and the distance we have to go.”
“But we don’t know the distance—CM-26 keeps moving.”
“Sure it does.” The boredom and frustration had been swept away by excitement and certainty. Rick suddenly had a clear mental picture in his head. “See, here’s what we do. We make two tables. The first has two columns. It shows times, say, every hour from the time we started, in the left column. The right column shows the distance from CM-2, where we started, to CM-26 for the corresponding time. It has to be a table, because CM-26 keeps moving. The information to make that distance table comes right out of the coordinates given to us in the problem packet.”
“But we don’t know the travel time—that’s what we’re supposed to find out!”
“I know. Let me finish. Now, we make another table. This one also has two columns. The left column is the same list of times, in hours. The right column says how far the ship goes in that much time, assuming that we accelerate for half the time, and decelerate for the other half. We already had the formula in class for the distance traveled in a given time with a given constant acceleration. It’s just half the time multiplied by the final speed.”
“But we don’t know the final speed!”
“Yes we do—at least, we know the formula for it. We had it on a test, it’s just the acceleration multiplied by the time.”
“If you say so. But I still don’t know what we do next.”
“We’re almost finished. We have two time/distance tables, right? Now we plot them both as curves on one graph. The first curve is the distance to CM-26, hour by hour. The second curve shows the ship’s distance traveled, also hour by hour. Those two curves cross somewhere—they have to, or we’d never be able to reach CM-26. And the time where they cross is the travel time—our answer. It’s the time when the ship will have gone just far enough to reach the position of CM-26. An approximation, but I bet it will be closer than ten percent.”
Alice was shaking her head. “You’ll have to go through all that again. You went too fast for me. Say it one more time.”
“I’ll do more than say it—we’ll do it, together. You make the table for the distance to CM-26. I’ll make the table for ship’s distance traveled. Then we compare and make the graph. Here’s a calculator for you.”
While Alice began to work out distances to CM-26, Rick began to make the table for ship’s distance. The result made him whistle in astonishment. A quarter of a gee didn’t sound like much of an acceleration, compared with the five gees and more that they had briefly endured on the way up to low Earth orbit. But keep it going, and the results were amazing. After an hour of accelerating and another of decelerating, you had traveled over thirty thousand kilometers. No big deal. Increase that to two days, though, and you were eighteen million kilometers away from your starting point. And in eight days. . .
“Alice, I want you to check this and see if I’m screwing up somewhere.”
“What’s wrong?” She had been working steadily, sitting on the folded bunk and biting her lip in concentration. Now she paused and smiled at Rick. For a change, the smile involved her whole face. Looking at her it was hard to believe that she usually seemed remote and uninvolved.
Rick held out his calculation. “According to this, if we travel for eight days we’ll have gone almost three hundred million kilometers. We’d be out in the Belt!”
“Let me look.” And, as she quietly did her own calculation. “You know, nobody said the trip out would take weeks and weeks—we just assumed it. Maybe that’s the point of the problem we were all given.”
“You mean, they want to see if we can figure out for ourselves that we’re going to be there sooner than we thought?”
“That’s right. You know what Turkey told us. The zingers wouldn’t stop when we left him.” She studied the final number on her calculator. “I get a different answer—twice as big as what you said.”
“Let me take a look.” Rick studied what she had done. “You weren’t listening to me. You’ve accelerated the ship for the whole eight days. You can’t do that. Look at your final speed, it’s nearly seventeen hundred kilometers a second. You’d go right through the Belt like—” Rick paused; like shit on skates was what he had been about to say, but he couldn’t see that expression pleasing Alice. “—like lightning,” he finished weakly. “You have to accelerate for half the time, then decelerate the other half, so you’re not going fast when you get there.”
“Let me do it again.” She repeated the calculation, slowly and carefully, and nodded. “I agree with you now. Four days accelerating at a quarter of a gee, then four days decelerating, takes you about two hundred and ninety million kilometers.”
“That could be more than enough to get you to Ceres.” Rick decided he sounded a bit know-it-all, and added, “I was just learning the asteroid distances when you came in.”
Instead of replying, Alice got up from the bunk, went across to the door, and locked it. She came back and stood directly in front of Rick. “You’ve finished the problem, everything but comparing the two tables.”