## Collisions

When two or more bodies approach each other, internal forces act between them, causing a variation in their linear momentum and in their energy, producing an exchange between them of both magnitudes. In this case, it is said that a crash or a collision has occurred between the bodies. For a collision to occur, it is not necessary that the bodies have been physically in contact in a microscopic sense; it is enough that they approach enough to have interaction between them.

Collisions or crashes occur very often in our daily life: a racket and a tennis ball; a bar and a ball; two billiard balls; a hammer and a nail…

The fundamental characteristic of a collision is that the forces which determine what happens during it, are only Internal forces (of interaction between the different bodies which crash). As a consequence of this, the speed of the center of mass of the system during the collision will be constant since the acceleration of the center of masses is produced only by the external forces acting on the system.

In a collision of two ordinary objets, both are deformed, with quite frequently, because of the large forces involved. When the collision occurs, the force goes from zero, at the time of contact, to a very high value, in a very short time. Next, return to zero violently.

According to Newton’s Second law, the net force on an object is equal to the rate of change of the momentum: Bioprofe | Collisions | 01

This equation applies to each of the objects that collide. If we multiply both sides by the time interval ∆t, we obtain: Bioprofe | Collisions | 02

We can see that the total change of the momentum is equal to the momentum. This concept helps a lot when it comes to forces that act during a short time interval, such as when a bat hits a baseball, where the force is not constant and its variation over time is usually a parabola of the type: Bioprofe | Collisions | 03

The average force Fm acts on a time interval ∆t. It communicates the same impulse (Fm∆t) as the real force.

## Conservation of energy and momentum in collisions

During most of the crashes, it is not known how the force of the shock varies as a function of time. However, much can be calculated about the movement after the shock, given the initial movement, by the laws of conservation of momentum and energy. When two objects collide, such as two billiard balls, the amount of movement is preserved.

If both objects are very hard and elastic, and no heat is produced in the shock, then the kinetic energy is also preserved. That is, the sum of the kinetic energy of the two objects is the same before and after the shock. Naturally, during the brief moment in which the two objects are in contact, a part of the energy, or all of it, is stored in the form of elastic potential energy. But if we compare the total kinetic energy before the shock, with after it, we will find that they are equal. Shocks in which the total kinetic energy is conserved are called “elastic shocks” or “elastic collisions”. Bioprofe | Collisions | 04

If we use the subscripts 1 and 2 to represent the two objects, we can write the kinetic energy conservation equation in the following way: Bioprofe | Collisions | 05

An elastic collision is an ideal case that almost never happens, because at least a small amount of thermal energy and, perhaps, another type of energy is produced during the collision. However, the collision of two hard and elastic balls, such as billiards, closely approximates the perfectly elastic case, and is ofter considered as such. Even when the kinetic energy is not conserved, the total energy of course is conserved.

Depending on the amount of kinetic energy that is lost, the collisions are divided into:

• Perfectly elastic collision: It is one in which kinetic energy is not dissipated and this is conserved.
• Inelastic collision: It is one in which part of the kinetic energy dissipates.
• Completely inelastic collision: It is one in which the maximum energy is dissipated. This máximum is not all the kinetic energy in the system, due to the conservation of the amount of movement requires that the system moves after the collision, and therefore conserves part of the kinetic energy.
Completely inelastic colisiones occur when the two partidles emerge and continue their march as a single particle whose mass is the sum of the two originals.
• Explosions: A process that is not strictly a collision, but can be treated mathematically as such. A single particle is broken down into two (or more) fragments, which happen to move separately. It would be the opposite of a completely inelastic collision. In this case, the kinetic energy of the system increases, usually at the expense of the internal energy of chemical origin.

In the case of frontal colisiones we can determine the value of the final velocities using the Coefficient of restitution, e, which was proposed by Newton: Bioprofe | Collisions | 06

Depending on the value obtained from this coefficient of restitution, we can know the type of collision: Bioprofe | Collisions | 07

Collisions in which kinetic energy is not conserved are called “inelastic crashes” or “inelastic collisions”. The kinetic energy that is lost is transformed int other forms of energy, often thermal energy, so that the total energy, as always, is conserved.

In this case we can affirm that: Bioprofe | Collisions | 08

## Elastic colisiones in one dimension

Suppose that, at the beginning, both partidles move with velocities v1 and v2 along the x-axis. After the collision, their speeds are v’1 and v’2. Bioprofe | Collisions | 09

For all v > 0, the particle mover to the right (increasing x), while for v < 0 the particle moves to the left (decreasing x).

According to the conservation of momentum, Bioprofe | Collisions | 10

As it is supposed that the shock is elastic, the kinetic energy is also preserved: Bioprofe | Collisions | 11

We have two equations, so we can solve two unknowns. If we know the masas and the initial velocities, we can clear the velocities after the collision, v’1 and v’2. First of all we must write the equation of conservation of momentum as: Bioprofe | Collisions | 12

And the corresponding to the kinetic energy as: Bioprofe | Collisions | 13

We divide the Second equation between the first one and finally we obtain: Bioprofe | Collisions | 14

For any frontal collision, the relativo velocity of the two partidles after the collision is the same as before, regardless of the masses.

## Elastic colisiones in two or three dimensions

The conservation of momentum and energy can also be applied to shocks in two or three dimensions and, in such cases, the vectorial nature of momentum is important. A frequent type of shock, not from the front, is one in which a particle moves (which we will call “projectile”) hits a second particle that is initially at rest (which we will call “white”). It is a common case in games like billiards and in atomic and nuclear physics experiments.

Next we can see a particle 1 (the projectile, m1) going through the x axis towards particle 2 (white, m2), which is initially at rest. If it is, we can say, billiard balls, m1, hits m2 and both are thrown off forming the angles ⍬1 and ⍬2, respectively, with respect to the initial direction of m1 (the x axis).

If it is electrically charged particles, or nuclear particles, they can begin to deviate before touching, due to the electriza or nuclear force, due to the electrical or nuclear force, between them. Bioprofe | Collisions | 15

Then we apply the laws of the conservation of the amount of movement and the kinetic energy to an elastic shock. According to the conservation of kinetic energy, and given that  v2=0, Bioprofe | Collisions | 16

We select the plane xy as the plane in which the initial and final movement quantities are. Since the momentum is a vector that is conserved, its components in the x and y directions remain constant.

In the x direction: Bioprofe | Collisions | 17

Due to initially there is no movement in the y direction, the y componen of the total momentum is zero, thus remaining: Bioprofe | Collisions | 18

## Inelastic collisions

Shocks in which kinetic energy is not conserved are called inelastic shocks or inelastic collisions. Some of the initial kinetic energy of these shocks is transformed into other types of energy, such as thermal or potential, so that the final total kinetic energy is less than the initial. The reverse case may also happen, in which potential energy is released, for example chemical or nuclear, and the final total kinetic energy is greater than the initial.

The common macroscopic shocks are inelastic, at least to some extent, and sometimes very inelastic. If two objects remain stuck as a result of a collision, the collision is said to be completely inelastic. Two plasticine balls which collide will stick together, like two railroad cars that snag when hit; these two are completely inelastic collisions. Not necessarily all the kinetic energy is transformed into other forms of energy when the collision is inelastic. Bioprofe | Collisions | 19

Even when the kinetic energy is not preserved in inelastic collisions, the total energy is retained, in the same way as the total amount of movement.