# Q & A: momentum, force, energy

Q:
Hello! Energy is a scalar, right? Now here is what I find confusing: when I push a mass into say +x direction, it will gain kinetic energy (that what force does) and it will accelerate to THAT direction (let's call it "to the right" in the lack of better name). However when I push it to the left, it will go to te left. How can this be if energy has no direction? what then makes the difference between left and right? There has got to be a differencem since two forces can cancel each other out and force is all about givin / taking energy.
- Anonymous
A:
Actually "force is all about giving/taking" momentum. Let me explain, then get to the energy part.

My favorite way to teach beginning mechanics is to start with the basic law that momentum is conserved. The momentum p of some mass m is p=mv, where m is the inertial* mass and the v is the velocity. the total momentum of a collection of masses m1, m2... is just given by the sum of their momenta, m1v1+m2v2.... The boldface symbols stand for vectors, things that point some way or other. Since p is conserved, it can't be changed but only traded between the masses. The force between two objects is by definition the rate at which momentum is being traded between them.  Thomas Moore wrote a nice introductory text that uses this approach, called "Unit C".

So how about force and energy? As you mentioned, an applied force can either increase or decrease something's energy. Or think of a ball twirling around on a string. The string pulls on the ball, changing its momentum (the direction it goes changes) but not changing its speed. So that force doesn't change the energy. There's no problem here really. Energy is also conserved, so it's traded between objects (and fields), but which way a particular force causes the energy to flow depends on how things are moving. It also depends on which reference from you choose. Form the point of view of somebody standing on a ball field, a good bunter reduces the kinetic energy of a pitched ball. From the point of view of somebody who was initially whizzing along with the pitch, the bunter gives the ball, initially at rest, a lot of kinetic energy. Both are perfectly legitimate descriptions.

Mike W.

*That means the same mass that appears in E=mc2, not the rest mass.

(published on 03/10/2011)