Kinetics of a Particle: Impulse and Momentum

Principle of Linear Impulse and Momentum

The equation of motion for a particle of mass m can be written as:

where a and ν are both measured from an inertial frame of reference. Rearranging the terms and integrating between the limits ν=ν₁ at t=t₁ and ν=ν₂ at t=t₂, we have:

or

This equation, which is referred to as the principle of linear impulse and momentum, provides a direct means of obtaining the particle's final velocity (ν₂) after a specified time period when the particles initial velocity is known and the forces acting on the particle are either constant or can be expressed as functions of time. Notice from the derivation that if (ν₂) is determined using the equation of motion, a two-step process is necessary; i.e. apply  ∑F=ma to obtain a, then integrate a=dv/dt to obtain ν₂. 

For problem solving, the equation will be rewritten in the form:

which states  that the initial momentum of the particle at t₁ plus the vector sum of all the impulse applied to the particle during the time interval t₁ or t₂ is equivalent to the final momentum of the particle at t₂. These three terms are illustrated graphically on the impulse and momentum diagrams shown below.

Each of these diagrams graphically accounts for all the vectors in the equation, 

two momentum diagrams are simply outlined shapes of the particle which indicate the direction and magnitude of the particle's initial and final momentum, mv₁=mv₂ ,respectively. Similar to the free-body diagram, the impulse diagram id an outlined shape of the particle showing all the impulses that act on the particle when it is located at the same intermediate point along its path. In general, whenever the magnitude or direction of a force varies, the impulse of the force is determined by the integration and represented on the impulse diagram as I=∫Fdt. If the force is constant for the time interval (t₁-t₂), the impulse applied to the particle is I=∫F_cdt(t₁-t₂), acting in the same direction  as F_c.

Linear Impulse

The integral of I=∫Fdt is defined as the linear impulse. This term is a vector quantity which measures the effect of a force during the time the force acts. The impulse vector acts in the same direction as the force, and its magnitude has units of force-time, e.g., N·s or lb·s. If the force is constant in magnitude and direction, the resulting impulse becomes:

which represents the shaded rectangular area as shown below.

Variable Impulse

Linear Momentum

Each of the two vectors of the form L=mv is defined as the linear momentum of the particle. Since (m) is scalar, the linear-momentum vector has the same direction as (v), and its magnitude (mv) has units of mass-velocity, e.g., kg·m/s and slug·ft/s.

Constant Impulse

Scalar Equations

If each of the vectors in resolved into its x, y and z components, we can write symbolically the following three scalars equations. These equations represent the principle of linear impulse and momentum for the particle in the x, y and z directions, respectively.

Procedure of Analysis

The principle of linear impulse and momentum is used to solve problems involving force, time and velocity, since these terms are involved in the formulation. For application it is suggested that the following procedure can be used.

Free-Body Diagram

Establish the x, y, z inertial frame of reference and draw the particle's free-body diagram in order to account for all the forces that produce impulses on the particle. The direction and sense of the particle's initial and final velocities should also be established. If any of these unknown, assume that the sense of its components is in the direction of the positive inertial coordinates. The velocities may be sketched near, but not on, the free-body diagram.

Principle of Impulse and Momentum

Apply the principle of linear impulse and momentum, 

If motion occurs in the x-y plane, the two scalar component equations can be formulated by either resolving the vector components of (F) from the free-body diagram.