**Branch :**First Year-Engineering Syllabus

**Subject :**Elements of Mechanical Engineering

## Linear Momentum

**Linear Momentum:**** **The product of the mass and velocity is called linear momentum of the particle.

Linear momentum **G**=m**v**

Linear monetum is a vector having the same direction as the velocity.

Thus, the rate of change of linear momentum is equal to the resultant force acting on the prticle.

Note that the direction of G and is not the same. In the component forms, we can write

where dot indicates differentiation with respect to time.

**Conservation of linear Momentum**: Since the time rate of change of linear momentum is equal to the resultant force acting on the particle, if there is no resultant force, the linear momentum is constant. This is the principle of conservation of momentum which is valid for the system of particles as well.

Suppose, there are two particles A and B that interact during an interval of time. If the interactive forces **F** and **-F** between them are the only unbalanced forces acting on the particles during the interval, it follows that the linear impulse on particle A is the negative of linear impulse on particle B. Therefore, the change in linear momentum of the particle A is the negative of the change in linear momentum of particle B.

Hence,

**Example:** Suppose a bullet of mass *m* strikes a block of mass M resting on a horizontal smooth floor and gets embeded into it. If the velocity of the bullet is *V*, find out the velocity of the(block bullet) after the bullet has embeded into it.

**Solution:** Applying the principle of momentum,

*mV=(M m)V _{f} *

where *V _{f}* is the final velocity.

Thus,

Now let us calculate the kinetic energy of the system before and after the impact.

Before impact,

(KE)_{1} = 1/2 *mV*^{2}

After impact

(KE)_{2} = 1/2 (*M m*) *V _{f}*

^{2}

Hence the change in kinetic energy

Thus, there is a loss of kinetic energy. This is because the force field is not conservative