Escape from the earth

Consider a stone thrown vertically from the earth’s surface with an initial vel­ocity. Because the weight of the stone, caused by the gravitational attraction between the stone and the earth, opposes the stone’s motion its velocity will be reduced with time (i. e. the stone has an acceleration g towards the earth). With a normal initial velocity gravitational acceleration will stop the stone at a certain height above the earth’s surface and it will then reverse direction and accelerate towards the earth, eventually arriving at its launch point with its launch velocity but in the reverse direction (if air resistance is neglected).

For moderate initial velocity it is sufficiently accurate to assume that the weight (and hence g) is constant. What happens, though, if the initial velocity is large enough for a great height to be reached before the stone’s direction of travel is reversed?

The weight of the stone, in accordance with the law of gravitation, is inversely proportional to the square of the distance between the masses. So supposing that the stone has a mass of 1 kilogram it will, at the surface of the earth, have a weight of 981 N. But what if it is moved away from the earth’s surface altogether? What if it is thrown upwards 1 kilometre, 100, 1000, 6000 kilometres? Let us pause here for a moment because the radius of the earth is not much more than 6000 km, 6370 km in fact, so at a distance of 6370 km from the earth’s surface, the force of attraction, i. e. the weight of the stone, being inversely proportional to the square of the distance from the centre of the earth – now doubled – will only be 1/4 of its weight at the earth’s surface; similarly, at 12 740 km it will be 1/9, at 19 110 km only 1/16, and so on (see Fig. 13.2). Notice that in the figure the distances are given from the centre of the earth, and not from the earth’s surface; for the earth is a very small thing in space, and if we are to understand the mechanics of space we must think more and more of the mass of the earth as concentrated at its centre.

The mass of the stone of course does not change, but as the weight changes so also does the acceleration (g) in proportion – this is just Newton’s Second Law again. So the rate at which the stone loses speed on the outward ‘flight’, though starting at 9.81 m/s2, gets less and less as the distance from the centre of the earth increases. This makes it more difficult to calculate how far the stone will go with a given starting velocity, but it has an even more interesting and important effect than this. For think of the stone returning to earth again; at great distances the rate at which it picks up speed will be very small, but the rate will increase until at the earth’s surface it reaches the definite and finite value of 9.81 m/s2. This, it will be noticed, is a maximum rate of increase, and it can be shown mathematically that even if the stone starts from what the mathematicians call infinity (which means so far away that it couldn’t come from any farther!) the velocity reached will also have a definite and finite maximum value, which is in fact 11.184 km/s (about 40 250 km/h). So, if a stone is ‘dropped’ onto the earth from infinity, it will hit the earth at 11.184 km/s; and, by the same token, if it is thrown vertically from the earth at 11.184 km/s it will travel to infinity – and never return. This velocity is called the escape velocity. If it is thrown with any velocity less than this, it will return.

What happens if it is thrown from the earth at a velocity greater than the escape velocity? Or is this not possible? Yes, it is not only possible, but in a sense it has been done though not quite in this simple way. And all that happens is that it still has a velocity away from the earth when it reaches infinity – and so will go beyond infinity – but since infinity is the limit of our imagination perhaps it will be best to leave it at that. The reader may have noticed that to simplify things, we have only considered the earth’s attraction on the stone, and the rest of the universe has been left out! But still the prin­ciple is illustrated.

It is important to remember that although the force of gravity, the weight of the stone, gets less and less as it travels farther and farther from the earth it never ceases altogether (at least not until the stone reaches infinity which is only another way of saying ‘never’). It is often stated, quite incorrectly, that ‘escape’ from the earth means getting away from the pull of the earth. This we

Fig 13.2 How weight varies with height R = radius of earth, i. e. approx. 670 km

can never do, the earth ‘pulls’ on all other bodies wherever they are – that after all, is the universal law of gravitation. Why then do astronauts talk about ‘weightlessness’? – and even demonstrate it? – we shall soon see.

In the meantime, it will be noticed that we have already introduced a new unit of velocity, the kilometre per second. Our reason for this is simply one of convenience; in this part of the subject we have to deal with very high speeds, and it is easier to remember these speeds, and even to think what they mean, as kilometres per second, than as so many thousands of knots, or metres per second. At the same time we must remember that our old friend g is still in m/s2, so if we wish to use any of the standard formulae of mechanics we must be careful to convert the velocities into metres per second.