Energy increases quadratically while force increases linearly. How is this possible if the kinetic energy is caused by force?

Question

Why doesn't energy rely on time, but only changes for an object in motion?

Say you are pushing an object as you walk, and your arm bone can exert 1,001 N (I made up this number) of force before breaking, using KE = 1/2mv^2 the energy exerted by and on it increases with speed if 1,000 N is constantly applied, but the force never changes. Does this mean that you can exert more energy on something without exerting more force on it, and having no change in stress levels? How or how not?

Also, similarly, since kinetic energy increases faster than force when pushing an object then what role does energy really play in it?

Answer 1

Yes. Power, which is energy supplied per unit time, is speed*force. So as the speed increases the pusher must supply more power to keep the force constant. That's why its easy to push a car 1mph but hard to push it 10mph. For a constant force accelerating a constant mass, the speed increases linearly with time. So the power increases linearly. If you take the time derivative of the kinetic energy that's the power required and if the acceleration, the dv/dt factor, is a constant then you see that it's proportional to v. "Kinetic energy increases faster than force" is meaningless. If the force is constant and the object moves then distance increases faster than force too. If you pushed on a tree and it didn't move would you say its speed increased slower than force?

Answer 2

Yes. Energy is not an "exerted" entity, its a measure. Imagine it as a sort of "indicator" of how much force you've put in over a distance. More precisely; we describe work as

$\vec{W} = \oint_c \vec{F} d\vec{l}$, which in this case is $\vec{W}=\vec{F}*\vec{x} = \Delta E$.

In this way, we see that the force can be constant, but as long as we continue to move the object, we can increase its energy. We can then describe energy as a function of time, as

$\vec{x} = \frac{\vec{F}}{m}t^2+\vec{v_0}t+x_0$, so

$E = \frac{\vec{F}^2}{m}t^2 +\vec{F}\vec{v_0}t + x_0$.

This only works if the force is constant, but as you can see, we CAN describe energy as a function of time, its just rarely worthwhile. The reason, conceptually speaking, why energy doesn't depend directly on time is that it is a measure of the "ability to do work" (at least at this level of physics). Thus it is dependant on the components of work.