EGR 280 Mechanics 13 – Impulse and Momentum of Particles

Impulse and Momentum Newton’s Second Law: vector relationship between force, acceleration and time. Work and Energy: scalar relationship between force, position and speed. Impulse and Momentum: vector relationship between force, velocity and time.

Consider a particle acted upon by a net force F. We have already seen that Where L = mv is the linear momentum of the particle. Write this expression as Ormv 1 + ∫Fdt = mv 2 The term ∫Fdt is called the linear impulse of F during the time interval [t 1,t 2 ]. If no external force is exerted on the particle, its linear momentum is conserved. If a force acts on the particle for a very short time, then ∫Fdt ~ F∆t and impulsive motion results: mv 1 + F∆t = mv 2

Direct Central Impact Consider two particles, A and B, moving to the right with known velocities v A and v B. If the two particles collide: m A v A + m B v B = m A v´ A + m B v´ B e = coefficient of restitution = (v´ B -v´ A )/(v A -v B ) = v´ B/A /v A/B vAvA vBvB u v´Av´A v´Bv´B Before collisionPoint of maximum deformation After collision

These two equations are then solved for the two unknown velocities after impact, v´ A and v´ B. The coefficient of restitution, e, can range from 0 (perfectly plastic impact, particles remain stuck together) to 1 (perfectly elastic impact, mechanical energy is also conserved). Oblique Central Impact If the velocities are not directed along the line of impact, then the impact is called oblique. Resolve the motion into components along the line of impact (the n direction) and perpendicular to the line of impact (the t direction). The velocity components in the n direction change as a result of the impact; the velocity components in the t direction do not change. n t vAvA vBvB