From Moon-fall to motions under inverse square laws

Abstract

The motion of two bodies, along a straight line, under the inverse square law of gravity is considered in detail, progressing from simpler cases to more complex ones: (a) one body fixed and one free, (b) both bodies free and identical mass, (c) both bodies free and different masses and (d) the inclusion of electrostatic forces for both bodies free and different masses. The equations of motion (EOM) are derived starting from Newton's second law or from conservation of energy. They are then reduced to dimensionless EOM using appropriate scales for time and distance. Solutions of the dimensionless EOM as well as the original EOM are given. The time interval for the bodies to fall is expressed as a function of the distance fallen. Formulae for the inverse were obtained. The coalescence times for the different cases are (a) where L is the initial separation of the two bodies and m1 is the mass of the fixed body, (b) and (c) where mT is the total mass of the two bodies and (d) where and is a measure of the ratio of the electrostatic force to gravity. The last formula may also be used when with the interpretation that there is no collision if t is infinity or imaginary. We also discuss this motion along the straight line as a special case of the general elliptic motion of two bodies. I believe that this paper will be useful to university tutors as well as undergraduate and even graduate students who prefer to consider the special case before the general case, and their relationship.

Comment on ‘From Moon-fall to motions under inverse square laws’

“…Correspondingly, the quantity representing distance in these equations, either the dimensionless x of equation ( 7) or the dimensional x of equation ( 9), cannot take the value of zero, as is implied by the computations leading to equation ( 8) on page 990 and in the steps performed on examples 2 and 3 on page 993 of the article.In fact, if one considers the interacting bodies as rigid spheres and representing by R T , R M and R S respectively the radius of the Earth, the Moon and the Sun, the lowest value that the dimensional x can take in equation ( 9) is x = (R T + R M ) (interaction Earth-Moon) or x = (R T + R S ) (interaction Earth-Sun). Moreover, the fact that x cannot be zero released the speed of the falling body, in equation ( 6), from taking an infinite value.Nevertheless, when the aforementioned corrections are implemented, the obtained numerical values are very close to the ones reported in the article [1]. Specifically, using the values R T = 6.378 × 10 3 Km, R M = 1.738 × 10 3 Km, R S = 6.960 × 10 5 Km, and the numerical values provided in Foong's paper at the beginning of section 2.6, page 993, the results for the dimensional time, using either equation ( 9) or equation ( 7) and (10) in the paper, are shown in table 1 (computed as if each numerical value has four significant figures): the computed relative percent error indicates that both values are practically the same.…”

“…Nevertheless, when the aforementioned corrections are implemented, the obtained numerical values are very close to the ones reported in the article [1]. Specifically, using the values R T = 6.378 × 10 3 Km, R M = 1.738 × 10 3 Km, R S = 6.960 × 10 5 Km, and the numerical values provided in Foong's paper at the beginning of section 2.6, page 993, the results for the dimensional time, using either equation ( 9) or equation ( 7) and (10) in the paper, are shown in table 1 (computed as if each numerical value has four significant figures): the computed relative percent error indicates that both values are practically the same.…”

“…Table 1. Results for dimensional time, computed from Foong's paper [1]. Finally, in the case of an object going through the centre of the attracting spherical body, the gravitational interaction inside the sphere needs to be taken into account.…”

“…J. Phys. 29 987).Reading with interest Foong's paper [1], we find it necessary to advise the interested and active instructor of physics that the dimensionless equation ( 7) or its equivalent dimensional equation ( 9) in the paper is only valid for the gravitational interaction between two point particles. In the case considering the gravitational interaction between spherical bodies as approximations of the Moon, the Earth and the Sun, the equations mentioned are only valid at points outside each mass distribution [2].…”

Nonclassical effects in geodesic motion

“…In mathematics, it would be very difficult to construct a local initial-wavepacket in spatial 3-dimensions (wherein the test particles were dropped), via the superposition of " hydrogen atomic eigenstates". Note that, even in classical mechanics, analytically solving the trajectory of free fall as a function of time is not trivial, which refers to the Inverse Kepler problem [34].…”