A new simulation . . Escape by Dr. J. Harris Chelsea Centre for Science Education University of London, England INTRODUCTION ln this module you are going to investigate how an object travels if it is launched vertically upwards from the earth's surface. You will be able to find out how far away it can travel, what initial velocity it must have to reach a certain height, and so on. Of course the computer doesn't actually do the experiments. It might - but in this case doesn't - simply tell you the results of trials which have been made. What it does do is to work out the results, based on particular physical laws, which we have good reason to have faith in. The calculations are also based on some simplifying assumptions. All this is explained in the body of the text. MOTION IN A GRAVITATIONAL FIELD Imagine that you are trying to throw a ball straight upwards, as high as you can. Obviously the harder you throw it the higher it will get before reaching its highest point and starting to fall back to earth. Q1 Suppose that for your strongest throw the stone gets to a height of 5m. How high would the stone get if you could give it twice the initial speed? A 7.1m B 10m C 14m D 20m or E 25m If you could answer that correctly you should know the height which the stone reaches depends on how much kinetic energy it has to start with, and that kinetic energy depends on (speed)^2. So twice the velocity means four times the kinetic energy. As the stone goes up its kinetic energy is transformed to potential energy. The potential energy depends on how high above the ground the stone is, and when the stone is 20m above ground, the potential energy is four times that for 5m above ground. So we can calculate how far the stone will go for a particular initial speed, however great. We can also calculate what initial speed it must be given to reach a particular distance. ***************************************************************** The Chelsea Science Simulation Project is the British counterpart to the Huntington Computer Project in the United States. In other words, it is a project to produce high-quality computer simulation modules consisting of a program, student's workbook, and teacher's guide. Most of the modules produced to date are in the areas of physics and biology and are in various states of test, revision, and final forms. A test version of the physics module, ESCAPE, is presented here. Future issues of Creative Computing will carry a complete article about the Chelsea Science Simulation Project and, hopefully, additional modules if enough readers Mant them. Write and let us know your interests. ***************************************************************** Q2 So far we have made at least two assumptions in this discussion. Do you know what they are? Suppose that a space probe were fired directly from the earth's surface, and fired fast enough to travel a great distance. Would the kind of argument used above still work? In particular, would the potential energy go on increasing uniformly with distance from the earth? You probably know that as one travels away from the earth the earth's pull, gravity, gets weaker. The force of gravity on an object follows the so-called inverse-square law. The radius of the earth is about 6,000km. The inverse-square law says that at twice this distance (l2,000km) from the earth's centre the force is only (1/2)^2 = 1/4 of what it is at the surface of the earth; at three times the distance the force is (1/3)2 = 1/9th, and so on. 1 1/4 1/9 1/16 Force on a kilogram (units of 9.8N) 1 2 3 4 Distance from centre of earth (units of 6,000km) The force pulling the object back towards earth steadily decreases as it gets further and further away. What about the potential energy? It goes on increasing - but does it still increase steadily, in proportion to the distance from the earth's centre? Theory says it wouldn't. It says that potential energy will increase more and more slowly as the probe gets further from the earth's centre and the gravity force gets weaker. The increase in potential energy in going from a distance 6,000km above the earth's surface to a distance 12,000km is less than the increase in going from the surface to a distance 6,000km. And again, the potential energy increase in going from 12,000 to 18,000 is less than the change in going from 6,000km to 12,000km, and so on. 1/2 1/6 1/12 proportional to increase in potential energy 1 2 3 4 distance (units of 6,000km) And so the kinetic energy is "used up" less and less quickly as the probe gets further away from earth. Now this raises a question. Would it be possible to give the probe so much kinetic energy that it would never be completely used up (transformed to potential energy)? lf so the probe would just go on indefinitely moving farther and farther away from earth. Perhaps it seems a reasonable idea that since the kinetic energy is transformed less and less quickly there will always be some left (if you eat half a cake today, then tomorrow eat half the remaining half to leave you a quarter, then eat half the quarter... you will, in theory anyway, always have some cake left). On the other hand however far from the earth the probe is (assume the earth is the only body in the Universe) there is always some gravitational force on it. And so you might argue that no matter how far away the probe is, nor how fast it is still moving, it will never be able to escape completely, never to return, because the earth will always be pulling it back, however weakly.