# Is my total work infinite?

Suppose I'm in space. I push a ball with a force. Then it will start to move and continue to move forever. So the distance traversed by the ball is infinite.

So we know, $$W=F × S =F × Infinity =Infinity.$$

That means that I had infinite energy in me and I transformed that energy into kinetic energy.

So isn't the energy in the world theoretically infinite?

• Are you continuing to push it forever? In which case, you would need an infinite amount of energy. If you are pushing it for a limited period and letting go then it might continue to move forever depending on where it was. For example, it might manage to go into orbit around something. The Moon is orbiting the Earth and the Earth is orbiting the Sun without any net work per orbit. This won't continue forever but it has been going on for a long time and probably will continue for another long time. – badjohn Apr 1 '18 at 18:23
• The $S$ in your equation is not how far the ball moves, it is how far the ball moves while you are pushing it. – Solomon Slow Apr 1 '18 at 19:56
• As @jameslarge noted, you cannot apply your formula for the "infinite" movement of your ball. After you pushed it there is no longer any force being acted on it. Then, your "W" will simply yield W=0xS= 0. So there is no extra work produced when the ball is drifting, only the work produced while you were pushing it. – Frotaur Apr 6 '18 at 15:42

## 2 Answers

Energy is the integral

$$W=\int_C \vec F\,d\vec s,$$

where $C$ is the trajectory, $d\vec s$ is infinitesimal displacement, and $\vec F$ is the force. If you don't know integral calculus, you can imagine this as the sum of small contributions of $F\Delta S$ along the trajectory.

Since you only briefly apply the force, it'll be nonzero only on a finite part of the trajectory $C$. After you stop applying the force, the integrand (i.e. the contributions $F\Delta S$) becomes zero for the remaining part of the trajectory, so further movement of the ball no longer contributes anything to the energy you calculate.

If your force is always in the same direction and is fixed in magnitude (while you're applying it), and you start with the ball at rest, then the energy will become

$$W=F\Delta x,$$

where $\Delta x$ is the length of the part of the trajectory where you were still applying the force.

• Thanks man. I don't know calculas ( I'm a fool), but still thanks. – Abu Safwan Md farhan Apr 1 '18 at 17:36
• @AbuSafwanMdfarhan please see if the edit makes the answer clearer. – Ruslan Apr 1 '18 at 17:42

You say...

$$W=F × S =F × Infinity =Infinity.$$

BZZZT! S is the distance that the force was applied, not the distance of the resulting movement. If you used your arms, S would be ~1 metre. If you think you can apply a force forever, think hard about that.