Momentum

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In classical mechanics, the momentum of a point particle is the mass m of the particle times its velocity v. Conventionally, momentum is indicated by the symbol p, so that

Both p and v are vectors. To distinguish p from angular momentum, it is often called linear momentum.

Conservation of momentum

Newton's second law states that the momentum of a particle changes in time when a force F acts on it,

where the acceleration a of the particle is introduced and it is assumed—as is common in classical mechanics—that the mass is constant (independent of time). Clearly, if no force acts on the particle:

which states that the momentum of a free particle (i.e., particle on which no force acts) is conserved.

Momentum of an N-particle system

The momentum of a system of N particles is the vector sum,

When the internal forces between the particles constituting the system satisfy Newton's third law (action = −reaction),

then

where we find on the right hand side the vector sum of external forces, Fexti, acting on the individual particles of the system. When the total external force is zero (either because all the individual external forces are zero, or because they sum vectorially to zero), then the total momentum of the system is conserved,

Application of conservation of momentum

Think of a rocket ship floating still in outer space. Assume that no gravitational, or other, forces are acting on it. The total momentum of the ship plus filled fuel tank is zero. Then ignite the rocket engine and assume that its exhaust gases go one way (say downward). The exhaust gases have mass and obtain velocity by the combustion, so that they have momentum, Pgas, directed downward. Because the total momentum is conserved (is zero), the ship gets momentum, Pship, upward,

so that the ship will get a velocity Vship upward.

Another example: suppose you are sitting in a driving car without seat belt. Your body gets the momentum: speed of vehicle, say 50 m/h, times your body weight. Suppose the car hits something and comes to a sudden stop (a strong force is acting on the body of the car and the car obtains zero speed). On you, however, no force is acting and your momentum will be conserved. Since your body weight does not change during the collision, your body will continue going forward with the same speed, 50 m/h. As the car has now speed zero, your body will move through the interior of the car with 50 m/h. It is no fun hitting the wind shield with this speed, so you better buckle on.