# Impulse

## Basic theoretical information

### Body impulse

**The impulse** (amount of motion) of a body is a physical vector quantity, which is a quantitative characteristic of the translational motion of bodies. Impulse denoted *p* . The impulse of the body is equal to the product of the mass of the body by its velocity, i.e. it is calculated by the formula:

The direction of the impulse vector coincides with the direction of the velocity vector of the body (directed tangentially to the trajectory). Impulse measurement unit – kg ∙ m / s.

**The total impulse of the system of bodies** is equal to the **vector** sum of impulses of all the bodies of the system:

**The change in the impulse of one body** is found by the formula (note that the difference between the final and initial impulses is vector):

where: *p *_{n} is the momentum of the body at the initial moment of time, *p *_{к} is the final moment . The main thing is not to confuse the last two concepts.

**Absolutely elastic impact** is an abstract impact model, which does not take into account energy losses due to friction, deformation, etc. No other interactions except direct contact are taken into account. With an absolutely elastic impact on a fixed surface, the velocity of the object after an impact on the module is equal to the velocity of the object before the impact, that is, the magnitude of the pulse does not change. It can only change its direction. The angle of incidence is equal to the angle of reflection.

**An absolutely inelastic blow** is a blow, as a result of which the bodies unite and continue their movement as a single body. For example, the plasticine ball, when dropped on any surface, completely stops its movement, when a collision of two cars triggers a clutch and they also continue to move on together.

### Momentum conservation law

During the interaction of bodies, the impulse of one body can be partially or completely transferred to another body. If external bodies from other bodies do not act on the system of bodies, such a system is called **closed** .

**In a closed system, the vector sum of the pulses of all bodies entering the system remains constant for any interactions of the bodies of this system among themselves. **This fundamental law of nature is called the **law of conservation of momentum (LES)** . The consequence of it are the laws of Newton. Newton’s second law in pulsed form can be written as follows:

As follows from this formula, if external forces do not act on the system of bodies, or the action of external forces is compensated (the resultant force is zero), then the change in momentum is zero, which means that the total momentum of the system is preserved:

Similarly, it is possible to reason for the equality to zero of the projection of force on the selected axis. If external forces do not act only along one of the axes, then the impulse projection onto this axis is saved, for example:

Similar records can be compiled for the remaining coordinate axes. One way or another, you need to understand that in this case the impulses themselves can change, but it is their sum that remains constant. In many cases, the law of conservation of momentum makes it possible to find the velocities of interacting bodies even when the values of the acting forces are unknown.

### Saving the projection pulse

There are situations when the law of conservation of momentum is only partially fulfilled, that is, only when designing on one axis. If a force acts on a body, then its impulse is not preserved. But you can always choose an axis so that the projection of force on this axis is zero. Then the projection of the momentum on this axis will be saved. As a rule, this axis is chosen along the surface on which the body moves.

### Multidimensional case Vector method

In cases where bodies do not move along one straight line, then in general, in order to apply the law of conservation of momentum, it is necessary to paint it on all coordinate axes involved in the problem. But the solution of such a problem can be greatly simplified by using the vector method. It is used if one of the bodies rests before or after the impact. Then the law of conservation of momentum is written in one of the following ways:

It follows from the rules of addition of vectors that the three vectors in these formulas must form a triangle. For triangles, the cosine theorem is applied.