Conservation of momentum

External and internal forces

A system is the set of bodies that are objects of study. Any action of an external agent on a body that constitutes the system is defined as an external force. The forces of interaction between the bodies that constitute the system are determined as internal forces. In a collision between two vehicles, for example, the furniture is considered as the system, and the forces generated from the collision are internal forces, existing through the interaction between the constituent objects of the system.

Impulse Theorem and Conservation of Momentum

Impulse (I) is the vector quantity that results from the product of the force applied to an object (external force) and the time of application of the force. Therefore, we can write:

I = F. Δt

The so -called Impulse Theorem shows that this quantity is equal to the change in momentum .

I = ΔQ

I = _FINAL Q – _{INITIAL Q}

A system is only considered conservative if there is no action of external forces , therefore, for a conservative system, the impulse is zero. With this in mind, we can write that the final momentum of a system must be exactly equal to its initial momentum.

I = _FINAL Q – _{INITIAL Q}

0 = _FINAL Q – _{INITIAL Q}

_FINAL Q = _{INITIAL Q}

From the understanding that the momentum of a system is conserved, many situations can be understood. An example is the possibility of determining the recoil speed of a weapon after firing a projectile.

Example

Imagine a 20 g projectile (20 x 10 ^{– 3} kg) that came out of a 5 kg shotgun with a velocity of 500 m/s. What would be the recoil speed of the shotgun?

We can understand that, initially, the shotgun+bullet system is stopped, so the initial momentum is zero. After firing, bullet and shotgun have opposite speeds. Taking the direction of the bullet as positive, we can write:

_FINAL Q = _{INITIAL Q}

Q _BULLE – Q _ESP = 0

The sign must be negative for the momentum of the shotgun, as it is in the opposite direction to that adopted as positive. Knowing that momentum is the product of mass and velocity, we write:

Q _BULLET – Q _ESP. = 0

Q _BULLE = Q _ESP

M _BULLE .V _BULLE = M _ESP . V _ESP

20 . 10 ^{– 3} . 500 = 5 . V _ESP

0.02 . 500 = 5 . V _ESP

10 = 5 . V _ESP

V _ESP = 2 m/s

Therefore, after firing, the recoil speed of the shotgun is 2 m/s (7.2 km/h).