The linear impulse-momentum relation

Linear momentum of a particle: The symbol L denotes the linear momentum and is defined as the mass times the velocity of a particle.

Example 1:

Impulse of a force from time t₁ to t₂: The integral of the force over the time interval of concern is its impulse. The impulse of a force is a vector given by the integral

Example 2:

Newton’s 2^nd law: Newton’s second law states that the resultant of all forces applied on a particle is equal to the rate of change of linear momentum of the particle.

This reduces to the more familiar statement of this law if one notes that in Newtonian mechanics it is assumed that mass is constant, and, therefore, .

The linear impulse-momentum relation: Integration of Newton’s 2^nd law over the time interval from t₁ to t₂ results in

Example 3:

Therefor, the linear momentum of a particle is changed by the impulse of the resultant force on the particle. There will be conservation of linear momentum only if the impulse of the resultant force is zero.

For a system of particles: Consider the system of particles shown below. Each particle in the system has a mass m_i and at time t₁ has a velocity v_i, and from time t₁ to time t₂ is acted upon by the resultant external force F_i and the internal forces of interaction between the particles of f_ij. As a result of the impulse of the internal and external forces on each particle, at time t₂ the particle with mass m_i has a velocity v^*_i.

Writing the impulse momentum equation for each particle and adding them up, considering that Newton’s 3^rd law requires that , we get for n particles

Therefore, the total linear momentum of the system is changed by the impulse of the external forces. Recalling the relation , we can rewrite this equation as

Example 4:

ã Mehrdad Negahban and the University of Nebraska, 1996-2002.

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Department of Engineering Mechanics, University of Nebraska, Lincoln, NE 68588-0526