Foremost among the laws governing collision is the conservation of linear momentum as there is no external force involved or the external force is small enough with respect to collision force and can be neglected without affecting the result in any appreciable manner.

For the system of colliding bodies, linear momentum of the system is same before and after collision :

For two colliding bodies, the above conservation law can be written as :

Conservation of linear momentum |
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Suffix "1" and "2" refer to the two colliding bodies, whereas suffix "i" and "f" refer to the "initial" and "final" states (before and after) of the collision respectively. It is important to visualize that two bodies will collide when their trajectory is such that the physical dimension of the bodies overlap. Further, one of the bodies approaches second body with greater speed (

Generally, the body with greater velocity that collides with another body is conventionally termed as "projectile" and the body which is hit is termed as "target".

We must understand that the law of conservation of linear momentum is valid for a closed system, wherein there is no exchange of mass between the system and its surrounding. Also, we must note that above mathematical construct is a vector equation. This can be expressed in component form as :

When bodies collide head-on, they move along a straight line. Such collision in one dimension can be handled with any of the component equations. For convenience, we can drop the component suffix and write the conservation law without suffix :

Collision in one dimension |
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As a reminder, we need to emphasize here that consideration of conservation of linear momentum is independent of following important aspects of collision :

- Collision force or time of collision
- Nature of collision : elastic or inelastic
- Dimension of collision : one, two or three

** Conservation of linear momentum in completely inelastic collision **

Completely inelastic collision is a special case of inelastic collision. In this case, the colliding body is completely embedded into target after collision. The two colliding bodies thereafter move with same velocity. Let the common speed be V such that :

Completely inelastic collision |
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Putting this in to the equation of linear momentum, we have :

If the collision is one dimensional also, then we can drop vector notation :

#### Example 1

Problem : A ballistic pendulum comprising of wooden block of 10 kg is hit by a bullet of 10 gm in horizontal direction. The bullet quickly comes to rest as it is embedded into the wooden block. If the wooden block and embedded bullet moves with a speed of 1.414 m/s after collision, then find the speed of bullet.

Ballistic pendulum |
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Solution : The collision lasts for short period. During collision, the combined mass of the block and bullet is subjected to gravitational and tension forces, which balance each other. As such, the system of colliding bodies is subjected to zero external force. The governing principle, in this case, is conservation of linear momentum. Now, as the bullet sticks to the wooden block, the collision is completely inelastic.

Let m and M be the masses of bullet and block respectively. Let us also consider that bullet strikes the block with a velocity "v" and the block and embedded bullet together move with the velocity "V" after the collision. We assume that bullet hits the block head on and the combined mass of block and bullet moves in the horizontal direction just after the collision. Hence, applying conservation of linear momentum in one dimension, we have :