# Energy, time and force

Consider force $F$ acting on a body of mass 1000kg and the displacement be $s$. So energy required to do so is $F$x $s$. Now consider the same force causing same displacement on body of mass 1kg. Here to energy required(according to equation) is same. But certainly more energy is required in the first case. How is this possible? Does it mean that no more energy is needed for prolonged application of force? This is the main reason I don't understand why work/energy is force times displacement .Instead force times time makes more sense to me.

You have to realize that the two bodies will reach a different final speed. Both will have the same kinetic energy (equal to the work of the force), but the larger one will have a considerably smaller speed.

The force is only applied for that distance, the fact that it is the same implies that the smaller object will have lower speed. Note that the relation does not imply the distance the object will move; instead, the displacement is the distance over which the force is applied. $$Fs=\frac{1}{2}mv^2$$ Hence, because the objects have the same energy but the mass of one is greater, they will have different velocities. This is where your energy "gap" comes in. But no more energy is required for the first compared to the second.

Your understanding of the fact that the first case needs more energy is flawed: they both need the same. In fact, the second case would need more power, because the time it takes is less. Keep in mind that work is the useful work done, and thus inefficiency is not accounted for.

When I was in high school, my teacher would use a mouse and an elephant for this kind of problem. Let us put a mouse and an elephant on ice skates.

You push the elephant, whose mass is $m_e$, with force $f$. The elephant slowly accelerates. You keep pushing until it eventually moves a distance $s$. By this time, the elephant is slowly moving at velocity $v_e$.

You push mouse, mass $m_m$, with the same force. The mouse quickly accelerates, and quickly reaches a distance $s$. The mouse is moving quickly with velocity $v_m$.

You did the same work on both, so both have the same kinetic energy.

$fs = \frac{1}2 m_e v_e^2 = \frac{1}2 m_m v_m^2$