EQUIVALENCE MECHANICAL WORK / KINETIC ENERGY

• On the usefulness of the equivalence between mechanical work and kinetic energy

n terminal ballistics, the effectiveness of a projectile in target is characterized by its kinetic energy. The latter gives information on the mechanical work that the projectile is capable of doing on impact, i.e. the damage or lesions that it is likely to cause to the target.

To understand the phenomenon, we must look at the forces that arise during the interaction between the projectile and the target. Indeed, physics tells us that in mechanics, the field that interests us here, the shape or the movement of a body can only be modified by the action of a force (or forces) and that, for there to be a force (or forces), there must be the interaction of at least two bodies. Let us note in passing that if there is a deformation of a body it is that there is a displacement of its constituent elements in relation to each other, thus, in fine, movement.

• The calculation

The second law of dynamics states that the sum of the forces fi applied to a body is equal to the product of its mass m and its acceleration a, i.e. the variation of its velocity with respect to time: a = dv/dt. In condensed mathematical language, this law is written as follows for any number n of forces :

Equation : 1

or, given that a = dv/dt :

Equation : 2

When several forces fi are applied to a body, we can make the vectorial sum of them. We thus obtain their resultant F i.e. :

Equation : 3

This gives the simplified expression :

Equation : 4

It is a differential equation (since we are working on small variations) of the first order (because the derivative intervenes only once). Its solution is not complicated.

We start by separating the variables :

Equation : 5

Let us stop for a moment on the above relation, it is interesting. It tells us that a force F, applied for a very short time dt, changes the velocity of a body of mass m by a small proportional amount dv. This is the relation describing the impulse of a force. Let us specify that this is not a mere thought. This formula is, or at least, has been applied in propulsion, in particular pulsoreactors.

Let's continue our way by asking ourselves the question : if we now know what a force applied during a very short time dt gives, what about the same force applied over a very short distance dx.

So we have to go from the time dt, to the distance dx traveled during the same period of time. Let's change the variabl

We know that a velocity is a distance traveled during a certain time, that is :

This gives us :

We finally obtain, by repeating the three steps of our calculation :

We obtain a new differential equation :

Equation : 6

Which is easily integrated. We obtain :

That is :

Equation : 7

Fx is a force applied over a distance x. It is called mechanical work. We have just established the relationship between mechanical work and kinetic energy. It is the relation that allows us to understand, in particular in wound ballistics, the mechanism of wounds formation.

• Application

For a given amount of kinetic energy at the moment of impact, equation 7 tells us that the smaller the distance x covered by the projectile in the target, the greater the force F and therefore the greater the damage or, in the case of wound ballistics, the greater the injuries.

The munitions manufacturer seeking anti-personal effectiveness therefore has interest in creating projectiles with strong slowing down in target by using mechanisms such as expansion or destabilization if the former is not possible.