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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
F
d ( mv )
dt
Where L = mv is the linear momentum of the particle.
Write this expression as

t2
t1
Or
Fdt  
( mv ) 2
( mv )1
d (mv )
mv1 + ∫Fdt = mv2
The term ∫Fdt is called the linear impulse of F during the time interval [t1,t2].
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:
mv1 + F∆t = mv2
Direct Central Impact
Consider two particles, A and B, moving to the right with known velocities vA
and vB. If the two particles collide:
vA
vB
Before collision
u
Point of maximum
deformation
v´A
v´B
After collision
mAvA + mBvB = mAv´A + mBv´B
e = coefficient of restitution = (v´B-v´A)/(vA-vB) = v´B/A/vA/B
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
vB
t
vA