The rate of change of momentum

As with conservation of energy, we need a way to measure and calculate the transfer of momentum into or out of a system when the system is not closed. In the case of energy, the answer was rather complicated, and entirely different techniques had to be used for measuring the transfer of mechanical energy (work) and the transfer of heat by conduction. For momentum, the situation is far simpler.

In the simplest case, the system consists of a single object acted on by a constant external force. Since it is only the object's velocity that can change, not its mass, the momentum transferred is

Δp = mΔv ,

which with the help of a = F/m and the constant-acceleration equation a = Δv/Δt becomes

Δp = maΔt

= FΔt .

Thus the rate of transfer of momentum, i.e., the number of kg·m/s absorbed per second, is simply the external force,

[relationship between the force on an object and the rate of change of its momentum; valid only if the force is constant]

This is just a restatement of Newton's second law, and in fact Newton originally stated it this way. As shown in figure k, the relationship between force and momentum is directly analogous to that between power and energy.

Power and force are the rates at which energy and momentum are transferred.

The situation is not materially altered for a system composed of many objects. There may be forces between the objects, but the internal forces cannot change the system's momentum. (If they did, then removing the external forces would result in a closed system that could change its own momentum, like the mythical man who could pull himself up by his own bootstraps. That would violate conservation of momentum.) The equation above becomes

[relationship between the total external force on a system and the rate of change of its total momentum; valid only if the force is constant]

Walking into a lamppost

This is also the principle of airbags in cars. The time required for the airbag to decelerate your head is fairly long, the time required for your face to travel 20 or 30 cm. Without an airbag, your face would hit the dashboard, and the time interval would be the much shorter time taken by your skull to move a couple of centimeters while your face compressed. Note that either way, the same amount of mechanical work has to be done on your head: enough to eliminate all its kinetic energy.

Ion drive for spacecraft

A toppling box