Question

the force of gravity between two objects is given by ( f_g=\frac{-gm_1m_2}{r^2} ), where ( g ) is the gravitational constant, ( m_1 ) and ( m_2 ) are the masses of the objects, and ( r ) is the distance between the objects centers. there is a vertical asymptote at ( r = 0 ). what does this mean in context of the problem? as the distance between the objects approaches 0, the gravitational force between them approaches infinity. as the masses of the objects approach 0, the gravitational force between them approaches 0. as the masses of the objects approach 0, the gravitational force between them approaches infinity. as the distance between the objects approaches 0, the gravitational force between them approaches 0.

Explanation:

A vertical asymptote at \(r = 0\) in the function \(F_g=\frac{-Gm_1m_2}{r^{2}}\) means that as \(r\) (the distance between the objects' centers) approaches \(0\), the value of \(F_g\) (the gravitational force) behaves in a particular way. For a rational function \(y = \frac{a}{x^{n}}\) (\(n>0\), \(a
eq0\)), as \(x\to0\), \(y\to\pm\infty\). In the context of the gravitational - force formula, since \(G\), \(m_1\), and \(m_2\) are non - zero ( \(G\) is a constant, \(m_1\) and \(m_2\) are masses of objects, so \(m_1>0\), \(m_2>0\)), when \(r\to0\), \(F_g=\frac{-Gm_1m_2}{r^{2}}\to-\infty\) (the negative sign indicates the direction of the force, but in terms of magnitude, \(\vert F_g\vert=\frac{Gm_1m_2}{r^{2}}\to\infty\)).

Answer:

As the distance between the objects approaches \(0\), the gravitational force between them approaches infinity.