Physics Problem: Binary Star System and Comet Speed

Question:

A binary star system has two stars, each with the same mass as our Sun, separated by d = 9.00 × 10¹⁴ m. A comet is very far away and essentially at rest. Slowly but surely, gravity pulls the comet toward the stars. Suppose the comet travels along a straight line that passes through the midpoint between the two stars.

What is the comet’s speed at the midpoint? (Express your answer with the appropriate units)

Answer and Explanation:

We begin by noting that if we take the gravitational potential energy to be zero at infinity, then the loss in potential energy as the comet moves from infinity to the midpoint is converted entirely into kinetic energy.

Each star has the mass of the Sun:

Distance between the stars:

So the distance from the comet (at midpoint) to each star is:

Gravitational potential energy per unit mass due to one star at distance r is:

Hence, total gravitational potential energy per unit mass from both stars at midpoint is:

Substitute r = d / 2 into the equation:

Change in energy from infinity to midpoint is:

This change in potential energy converts into kinetic energy at the midpoint:

Now substitute values:

First calculate the numerator:

Then divide by denominator:

Take the square root: