Example: The Dog
Let's take an example. You are standing still when your dog runs up and jumps on you. Together, you and the dog fall backwards. Your mass is 40 kg and the dog is 20 kg. The dog is coming at you at 2 m/s. Since linear momentum is conserved in this situation, the total linear momentum before the collision must equal the total linear momentum afterwards.
The linear momentum before the collision is
You and the dog are basically stuck together as you are falling. The two of you are moving with the same velocity backwards. The linear momentum after the collisions is then
Setting these expressions equal and solving yields the velocity at which you and the dog will be moving backwards.
After the collision, you and the dog will be moving at 2/3 m/s backwards.
There is an interaction between you and the dog during the collision that causes you to start moving and the dog to slow down. When the dog hits you, he pushes on you and exerts a force on you. You start moving and gain some linear momentum. At the same time, you are exerting a force on the dog to slow him down. The force the dog exerts on you is exactly equal and opposite to the force you exert on the dog. Nothing influences you or the dog from the outside so linear momentum is said to be conserved.