What is energy?

A number, more precisely a scalar, that is conserved in a closed and isolated system throughout time. It can seem counter intuitive since time is a measure of increasing entropy where our conserved number, energy, is broken down into the smallest possible amount of energy given certain force constraints and boundary conditions, the energy ground state. We can also think of kinetic energy as the double integral of the force. Since a force is defined as F = mA and acceleration A is equal to dv/dt. Back from kinematics we know that when you integrate an acceleration you get a velocity and integrating velocity again gives us a (1/2)v2 (integrating by dv of course). Therefore the double integral of F = mA (with respect to time and then with respect to velocity) gives us kinetic energy 1/2mV2. Integrating only once gives us the momentum P=mV, this is why the force equation is often written as F = dp/dt too. You might object saying this cannot be so and that the integral of velocity is spatial since dx/dt = v. True, but think of it as a definite integral from when the force was applied to when the force stopped being applied leaving the object with a final velocity of V and a total energy of 1/2mV2. This velocity integral is the total distance traveled over an infinitesimal delta t. This should be the same as saying that the force applied over a distance X is the energy and that is why the [potential energy](www.theweinerworks.com/?p=1563) is the amount of distance changed due to a force acting on the object. The fun part is considering the case of two objects which have the same velocity vectors. Like two arrows fired one after the other. The two arrows traveling with the same velocity do not see each other with different velocities relative to each other. In the arrow frame of reference the arrows appear to have no kinetic energy, but instead it is the archer and target frames of reference with all the kinetic energy. This craziness is why people give up studying relativity. Remember, that the force integral is actually the impulse so you could say that the arrows were the objects that experienced the jerk which is a discrete instantaneous change in acceleration. But that would ignore the fact that there was an equal and opposite force (acceleration) applied to the archer. The only satisfactory answer for me is that the force was enough to overcome the inertia of the arrow and not the inertia of the archer. This means that the arrow experienced a non-inertial frame for a small amount of time and the archer stayed in his (or her)inertial frame. We have another number called the mass of an object which is the amount of resistance an object has to changing it's inertial frame.

/r/askscience Thread