Section C: Impulse & Momentum Section 3.8 – Introduction Work/Energy was obtained by integrating the Equation of Motion with respect to displacement over which the force is applied. We will now consider the integral of the Equation of Motion with respect to the time interval over which the force is applied. This will lead to the equations of Impulse & Momentum.

The Equations of Impulse and Momentum are especially useful in situations where the forces only persist for a brief duration. e.g. Impact problems.

Section 3.9 – Linear Impulse & Linear Momentum Consider the curvilinear motion of a particle. The position vector r is measured from some convenient fixed reference point, O. The velocity vector of the particle is: and is tangent to the path of the particle. The resultant force F of all forces acting on the particle of mass m is in the direction of

We can write the Equation of Motion as: Where G ≡ mv is the Linear Momentum of the particle. The units of G are kgm/s is a vector equation. We still require that mass is constant.

We can rewrite the equation of motion in component form, for example in rectangular coordinates we have: These are 3 independent scalar equations.

Linear Impulse – Momentum Principle All that we have done is to rewrite the Equation of Motion in terms of the Linear Momentum. We can now, however, find the effect of F on G during an interval of time by integrating as follows: Linear Impulse of the particle.

The total linear impulse on the particle of mass m is equal to the corresponding change in the linear momentum of the particle. The Impulse Integral is a vector! We may write it in component form:

The three scalar component equations are independent. If F is itself a function of time F(t) then we must know F(t) either analytically of graphically in order to perform the integration.

Conservation of Linear Momentum If F =0 during some time interval then G is constant i.e. Linear Momentum is Conserved. Linear Momentum may be conserved in one direction, but not in another direction during the same interval of time.

If two particles interact during a time interval and if the interaction forces between the 2 particles are the only unbalanced forces acting on the particles during the interval then: This is the Principle of Conservation of Linear Momentum

Example Problem

Example Problem