The force of gravity between two objects is given by [tex]F_g=\frac{-G m_1 m_2}{r^2}[/tex], where [tex]G[/tex] is the gravitational constant, [tex]m_1[/tex] and [tex]m_2[/tex] are the masses of the objects, and [tex]r[/tex] is the distance between the objects' centers. Find the vertical asymptote of the graph of the function and explain its meaning in context.
Answer (2)
The vertical asymptote of the gravitational force function F g = r 2 − G m 1 m 2 is at r = 0 . As two objects come closer together (approaching a distance of zero), the gravitational force approaches infinity. However, this is a theoretical limit as real objects cannot occupy the same space.
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The problem provides the gravitational force formula F g = r 2 − G m 1 m 2 .
A vertical asymptote occurs when the denominator of the function is zero, which is when r 2 = 0 .
Solving for r , we find the vertical asymptote at r = 0 .
In context, this means the force of gravity approaches infinity as the distance between objects approaches zero, though the formula's validity diminishes at such small distances. The final answer is r = 0 .
Explanation
Understanding the Problem We are given the formula for the force of gravity between two objects: F g = r 2 − G m 1 m 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. We need to find the vertical asymptote of the graph of this function and explain its meaning in context.
Identifying the Vertical Asymptote A vertical asymptote occurs when the function approaches infinity (or negative infinity) as the input variable approaches a certain value. In this case, the input variable is r , the distance between the objects. The function F g will approach infinity when the denominator r 2 approaches zero.
Solving for the Asymptote To find the vertical asymptote, we set the denominator equal to zero and solve for r : r 2 = 0 r = 0 Thus, the vertical asymptote is at r = 0 .
Interpreting the Asymptote in Context In the context of gravity, the vertical asymptote at r = 0 means that as the distance between the two objects approaches zero, the force of gravity approaches infinity. However, this is a theoretical limit. In reality, the equation F g = r 2 − G m 1 m 2 is not valid when the distance between the objects approaches zero, because the objects cannot occupy the same space. Also, at very small distances, other forces (such as nuclear forces) become significant and the gravitational force is no longer the dominant force.
Final Answer The vertical asymptote of the graph of the function F g = r 2 − G m 1 m 2 is r = 0 . This means that as the distance between the objects approaches zero, the force of gravity approaches infinity, but the equation is not valid at extremely small distances.
Examples
Understanding vertical asymptotes is crucial in various fields, such as physics and engineering. For instance, when designing structures, engineers must consider how forces behave as distances approach critical points. In the case of gravitational forces, knowing the behavior near r = 0 helps in understanding the limitations of the classical gravity model and the need for more advanced theories at very small distances. This concept also applies to other force laws, like electrostatic forces, where similar asymptotes exist and require careful consideration in practical applications.
