Question

Two point-like particles, each of mass \(m\), are originally separated by a distance \(d\) and moving in opposite directions each with a speed of \(v\). What is the maximum value for \(v\) so that the particles will eventually move back together under the influence of the mutual gravitational attraction?

Short answer

The maximum initial speed of the particles so they move back together under mutual gravitational pull is \(v = \sqrt{2 G m / d}\).

Step-by-step solution

Gravitational force and escape velocity

Begin with the force of gravity formula \(F = G m^2 / d^2\) and the escape velocity formula \(v = \sqrt{2 G m / d}\). But as the particles barely escape each other's gravitational pull, their speed would be just under the escape velocity. Thus, maximum \(v = \sqrt{2 G m / d}\), but apply a minus to denote the particles eventually return.

Calculating the Energy

The total energy of the system before and after must be equal due to the Law of Conservation of Energy. Initially, both particles have kinetic energy and potential energy. Total Kinetic Energy would be \(2 \times \frac{1}{2} m v^2 = m v^2\). Total Potential Energy would be \(- G m^2 / d\). After they've moved, their potential energy is zero at maximum extent.

Equating the Energies

So equate the initial total energy and the final total energy to get: \(m v^2 - G m^2 / d = 0\). Solve this equation for \(v\) to get the maximum value of \(v\) under which the particles will eventually move back together.

Key concepts

Escape Velocity

Imagine throwing a ball upwards; it eventually comes back down due to Earth's gravity. Now envision throwing it so fast that it never comes back down—that speed is what we call the escape velocity. It's the minimum velocity an object must have to break free from the gravitational pull of a celestial body, without further propulsion.

In our exercise, two particles are moving apart, but to find out if they'll come back together, we need to consider if they've reached escape velocity. If the particles move slower than escape velocity, they are bound to return towards each other due to gravity. The formula for escape velocity is derived from balancing gravitational potential energy with kinetic energy and is given by: \[v_e = \sqrt{2 G m / d}\], where \(v_e\) is the escape velocity, \(G\) is the gravitational constant, \(m\) is the mass of the particle, and \(d\) is the distance between the particles.

Gravitational Force

Gravitational force is the attractive pull between any two masses. It's a fundamental force of nature that keeps planets in orbit and binds galaxies together. For two point-like particles, this force can be expressed with Newton's law of universal gravitation: \[F = G \frac{m^2}{d^2}\], where \(F\) stands for the gravitational force, \(G\) is the gravitational constant, \(m\) is the mass of each particle, and \(d\) is the separation distance.

In our problem, the gravitational attraction between the two particles acts as the force that might eventually bring them back together, if their initial speed is not greater than the escape velocity.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed in an isolated system; it can only be transformed from one form to another. When we consider our particles, we've to factor in both kinetic and potential energy to understand the motion.

This principle ensures that the total energy at the beginning is the same as the total energy at the end. If we add up all forms of energy at the start (while the two particles are moving but still bound by gravity) and equate it with the sum of energies when they are at their maximum separation (where the potential energy is zero if they were to separate infinitely), we can use this law to solve for the maximum value of \(v\).

Kinetic Energy

Kinetic energy is the energy an object possesses because of its motion. It's calculated with the formula \[KE = \frac{1}{2} m v^2\], with \(KE\) representing kinetic energy, \(m\) the mass of the object, and \(v\) the velocity.

For our two particles, each has kinetic energy due to their velocities. The total kinetic energy is doubled since we have two particles. This energy contributes to their motion and, when considering their fate in gravitational terms, is balanced by their potential energy to determine whether they will escape each other's pull or eventually reunite.

Potential Energy

Potential energy is the stored energy of an object due to its position or state. In the context of gravity, gravitational potential energy is what an object has when it's within a gravitational field. The formula for this energy between two masses is \[PE = - G \frac{m^2}{d}\], where \(PE\) represents potential energy, \(G\) is the gravitational constant, \(m\) the mass, and \(d\) the separation distance.

The negative sign indicates that the force of gravity is attractive. In our exercise, as the particles start to move apart, their potential energy decreases and their kinetic energy increases until they reach a point where they cannot escape each other's gravitational pull, provided the speed is less than the escape velocity. At this stage, their total potential energy would be at its minimum value.