Elements of ballistic calculation for spacecraft gravity assist

The purpose of the study is an analytical description of the section of the ballistic trajectory corresponding to the normal fall of the spacecraft on the surface of an atmosphereless planet. In this case, the motion of a normally falling body is characterized by an increasing acceleration of gravity. The problem of the speed, time and acceleration of the normal fall of a body on the planet's surface in the absence of an atmosphere is reduced to solving a second-order differential equation, which is solved by the standard method. A feature of the solution is the formal use of the tabular integral at an intermediate stage. It turned out, however, that his formula is unreliable, namely, the derivative of the right-hand side is not equal to the integrand. It follows from this that the possible existing solutions to this problem, based on the use of the indicated tabular integral, are incorrect. The article presents the correction of this tabular integral, which is an incidental result of the study. In this work, the time equation of motion of a body normally falling on the surface of the planet in the absence of an atmosphere, as well as the time equations of its speed and acceleration are obtained. The results obtained can be useful in calculating passive gravity assist during interplanetary flights and calculating the sheer fall of small celestial bodies and spent structural element s of spacecraft.