Consider a universe that is shaped like a parabolic bowl, where the height of the bowl, h, is a function of the distance from the center, r. The height function can be expressed as h(r) = -a * r^2 + b, where a and b are positive constants.

One day, a mischievous cosmic entity decides to place a tiny ball at the edge of the universe. The entity sets the ball in motion, causing it to roll down the inner surface of the parabolic bowl, towards the center of the universe.

The cosmic joke lies in the fact that the ball will never actually reach the center. Instead, it will approach the center indefinitely, forever trapped in its futile journey.

To illustrate this joke, we can calculate the time it takes for the ball to travel a certain distance towards the center, as a function of its initial position.

  Assume that the ball rolls without friction or air resistance.
  The gravitational force acting on the ball is constant, with magnitude g.
  The ball starts at the edge of the universe, r = r_0, where r_0 > 0.

Problem:

   Determine the gravitational potential energy, U(r), of the ball as a function of its distance from the center, r.
   Calculate the kinetic energy, K(r), of the ball as a function of its velocity, v(r).
Use the conservation of energy to derive a differential equation that relates the ball's position, r(t), and velocity, v(t), as a function of time, t.
Solve the differential equation to find the time it takes for the ball to travel from its initial position, r = r_0, to a point closer to the center, r = r_f, where r_f < r_0.
Show that the ball's velocity approaches infinity as it approaches the center, and explain why the ball will never actually reach the center.

This problem captures the essence of calculus by using the concepts of derivatives, integrals, differential equations, and limits, while also alluding to the idea of existence being a cosmic joke, where our pursuits may be endless and ultimately futile.
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