We use Figure, conservation of energy, to find the distance at which kinetic energy is zero. When its speed reaches zero, it is at its maximum distance from the Sun. The object has initial kinetic and potential energies that we can calculate. If an object had this speed at the distance of Earth’s orbit, but was headed directly away from the Sun, how far would it travel before coming to rest? Ignore the gravitational effects of any other bodies. As we see in the next section, that is the tangential speed needed to stay in circular orbit. We noted that Earth already has an orbital speed of 30 km/s. Let’s consider the preceding example again, where we calculated the escape speed from Earth and the Sun, starting from Earth’s orbit. For real objects, direction is important. We examine tidal effects in Tidal Forces.) Neither positive nor negative total energy precludes finite-sized masses from colliding. (Even for greater values of r, but near the sum of the radii, gravitational tidal forces could create significant effects if both objects are planet sized. If r becomes less than this sum, then the objects collide. They apply to finite-sized, spherically symmetric objects as well, provided that the value for r in Figure is always greater than the sum of the radii of the two objects. Strictly speaking, Figure and Figure apply for point objects. Earlier we stated that if the total energy is zero or greater, the object escapes. It is possible to have a gravitationally bound system where the masses do not “fall together,” but maintain an orbital motion about each other. Energy is a scalar quantity and hence Figure is a scalar equation-the direction of the velocity plays no role in conservation of energy. What is remarkable is that the result applies for any velocity. We have simplified this discussion by assuming that the object was headed directly away from the planet. The object can never exceed this finite distance from M, since to do so would require the kinetic energy to become negative, which is not possible. On the other hand, if the total energy is negative, then the kinetic energy must reach zero at some finite value of r, where U is negative and equal to the total energy. When the total energy is zero or greater, then we say that m is not gravitationally bound to M. If the total energy is positive, then kinetic energy remains at r=\infty and certainly m does not return. Hence, m comes to rest infinitely far away from M. If the total energy is zero, then as m reaches a value of r that approaches infinity, U becomes zero and so must the kinetic energy. In Potential Energy and Conservation of Energy, we showed that the change in gravitational potential energy near Earth’s surface is \Delta U=mg(\,r\to \infty. Potential energy is particularly useful for forces that change with position, as the gravitational force does over large distances. The usefulness of those definitions is the ease with which we can solve many problems using conservation of energy. We defined work and potential energy in Work and Kinetic Energy and Potential Energy and Conservation of Energy. Gravitational Potential Energy beyond Earth
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