(33) is a vector equation, while energy is a scalar quantity. In this section, we are going to prove that the two-body equation of motion, Eq. Application to Two-Body Relative Motion # Where \(a_x\), \(a_y\), and \(a_z\) are the relative acceleration components. Since the total mechanical energy, \(E\), is a constant, its derivative with respect to time must be equal to zero: Now, let’s define each of the forms of mechanical energy a little more rigorously. You can examine the results from Two-Body Problem in the Co-Moving Frame to verify this is correct. Likewise, the point of lowest speed is at the farthest point on the orbit. The point of highest speed is at the point of closest approach. Gravity is a conservative force, which means that any change of kinetic energy is associated with an equal and opposite equal change of potential energy.Īs kinetic energy increases and the speed of \(m_2\) increases, the potential energy must decrease and \(m_2\) gets closer to \(m_1\). The potential energy is due to the location of \(m_2\) in the gravitational field of \(m_1\), while the kinetic energy is due to the velocity of \(m_2\) relative to \(m_1\). Non-conservative forces, such as friction and drag, result in a conversion of mechanical energy into heat. Where \(E\) is the total mechanical energy of the system and must be a constant.Ĭonservative forces allow infinite conversion of energy between forms. Planetary Depahture for Interplanetary Transfer.Review of Vectors, Kinematics, and Newtonian Mechanics
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