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kinetic energy paradox?



Hi.

Here's something I've been mulling over. When I find myself in a paradoxical situation it's usually because I've reasoned something incorrectly. I'm kinda stuck this time.

Consider a ball bearing rolling across a frictionless (yes, I know there must be a minimal friction force or the thing would never roll...) tabletop or desktop parallel to the edge of the table or desk. It's rolling with a constant speed, and therefore a constant kinetic energy. Suppose at a given instant, you give the ball bearing an impulse directed perpendicular to it's initial velocity at that instant. The ball bearing's parallel velocity component will not change, but there is now a non-zero prependicular velocity component. The magnitude of the total velocity is now different, and therefore the ball bearing's kinetic is now also different; it's increased.

As a result of the new non-zero perpendicular velocity, the total velocity is now pointing in a slightly different direction (the smaller the duration of application of the force the smaller the change in direction, right?).

Now, if you look at the gravitational force on Earth by the Sun, and assume Earth's orbit is circular, the Sun's gravitational force does no work on Earth and Earth's kinetic energy is constant. The gravitational force only serves to change the direction of Earth's velocity.

My quandry is this: why is Earth's kinetic energy conserved and the ball bearing's kinetic energy NOT conserved? In both cases, an impulse is imparted perpendicular to the velocity.

Is the answer just that Earth receives MANY such impulses continuously, one right after the other, and each impulse is *instantaneously perpendicular to the displacement (and velocity) vector at that same instant*? And it's the *tangential* speed that determines the orbital kinetic energy, and this component isn't changed.

I'm working on a unit on Newtonian dynamics and kinematics to use in an introductory astronomy course. Basic dynamics is too often neglected in the intro astro course, and yet we expect students to understand Kepler's laws and fundamental properties of orbital motion.


Cheers,
Joe

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