Chapter 15: Kinetics of a Particle: Impulse and Momentum
15.1 Principle of Linear Impulse and Momentum
 The equation of motion for a particle of mass m can be written as
F
 ma  m
dv
dt
(1)
where a and v are both measured from an inertial frame of reference.
 Rearranging the terms of equation (1) and integrating, we have

t2
t1
v2
F dt  m dv
v1
or

t2
t1
F dt  mv 2  mv1
(2)
 Equation (2) is referred to as the principle of linear impulse and momentum. It can be used to
solve problems involving force, time, and velocity.
 Each of the two vectors of the form L = mv in equation (2) is referred to as the particle’s
linear momentum. It has the same direction as v, and its magnitude mv has units of kg.m/s or
slug.ft/s.
 The integral I = ∫F dt in equation (2) is referred to as the linear impulse. The term is a vector
quantity which measures the effect of a force during the time the force acts. It acts in the
same direction as the force and its magnitude has units of N.s or Ib.s.
 The linear impulse can be interpreted as the area under the force versus time curve (see Fig.
15-1). When the force is constant in both magnitude and direction, the resulting impulse
becomes I 

t2
t1
Fc dt  Fc t 2  t1  .
 For problem solving, equation (2) can be rewritten as
mv1 

t2
t1
F dt  mv 2
(3)
 Equation (3) states that the initial momentum of the particle at t1 plus the sum of all the
impulses applied to the particle from t1 to t2 is equivalent to the final momentum of the
particle at t2. The three terms are shown in the impulse and momentum diagrams in Fig. 15-3.
 The vector equation (3) can be resolved into its x, y, z components to obtain the following
three scalar equations:
mv x 1 

t2
mv y 1 

t2
mv z 1 

t2
Fx dt  mv x 2
t1
t1
Fy dt  mv y 2
t1
Fz dt  mv z 2
(4)
15.2 Principle of Linear Impulse and Momentum for a System of Particles
 The principle of linear impulse and momentum foe a system of particles (such as the one
shown in Fig. 15-7) can be written as
 m v 
i
i 1


t2
t1
Fi dt 
 m v 
i
i 2
(5)
where Fi represents the external forces acting on the system of particles. Note that the
internal forces acting between particles do not appear in equation (5) because they occur in
equal but opposite pairs and therefore their impulses cancel out.
 Equation (5) can also be written as
mv G 1 

t2
t1
F dt  mv G 2
(6)
where m = Σmi and vG is the velocity of the center of mass of the system of particles.
15.3 Conservation of Linear Momentum for a System of Particles
 When the sum of the external impulses acting on a system of particles is zero, equation (5)
reduces to
 m v 
i
i 1

 m v 
i
i 2
(7)
 Equation (7) is referred to as the conservation of linear momentum. It states that the total
linear momentum for a system of particles remains constant during the time period t1 to t2.
 Substituting mvG = Σmivi into equation (7), we can also write
(vG)1 = (vG)2
(8)
This indicates that the velocity vG of the mass center for the system of particles does not
change when no external impulses are applied to the system.
 The conservation of linear momentum is often applied when particles collide or interact.
 When the motion of particles is studied over a very short time, the external forces may be
classified as impulsive or nonimpulsive.
o Nonimpulsive forces are forces that are so small their impulses can be considered to be
negligible in the impulse-momentum analysis. Examples may include the weight of a
body, the spring force from a slightly deformed spring having a relatively small stiffness.
o Impulsive forces are very large and produce significant change in momentum. They
cannot be neglected in the impulse-momentum analysis. Impulsive forces normally occur
due to an explosion or the striking of one body against another.
 Generally, the impulse-momentum analysis of a system of particles involves two stages:
1.
Apply the conservation of linear momentum (equation 7) on the system of particles.
2.
Isolate (i.e. draw the free-body diagram of) a particle and apply the principle of linear
impulse and momentum (equation 3) to obtain the internal impulse acting on that
particle.
15.4 Impact
 Impact occurs when two bodies collide with each other during a very short period of time,
causing relatively large (impulsive forces) to be exerted between the bodies.
 In general, there are two types of impact: central impact and oblique impact.
o Central impact occurs when the direction of motion of the mass centers of the two
colliding particles is along the line of impact. [The line of impact is a line passing through
the mass centers of the particles.]
o Oblique impact occurs when the motion of one or both of the particles is at an angle with
the line of impact (see Fig. 15-13).
Central Impact
 When two smooth particles are involved in a central impact, the following sequence of
events occurs (see Fig. 15-14):
 The particles undergo a period of deformation as they exert equal but opposite impulse on
each other.
 At the instant of maximum deformation both particles move with a common velocity.
 Afterward, a period of restitution occurs, in which case the particles will either return to
their original shape or remain permanently deformed. The equal but opposite restitution
impulse pushes the particles apart from one another.
 The ratio of the restitution impulse to the deformation impulse is called the coefficient of
restitution, e.
 By applying the principle of impulse and momentum to the particles during the deformation
and restitution processes, the coefficient of restitution is obtained as:
e
v B 2  v A 2
v A 1  v B 1
(9)
i.e. e is equal to the ratio of the relative velocity of the particles’ separation just after
impact to the relative velocity of the particles’ approach just before impact.
 In general, e has a value between 0 and 1.
o If e = 1, the collision is said to be perfectly elastic.
o If e = 0, the collision is said to be inelastic or plastic. Here, there is no restitution impulse.
The particles stick together and move with a common velocity.
 Note that the work and energy principle cannot be used for the analysis of impact problems
since it is not possible to know how the internal forces of deformation and restitution vary
during the collision. The energy loss during collision is the difference in the kinetic energies
of the particles before and after collision.
Oblique Impact
 When oblique impact occurs between two smooth particles, the magnitudes and directions of
the final velocities are unknown (see Fig. 15-15). Use the following procedure to obtain 4
equations for the 4 unknowns:
o Apply conservation of momentum on the system along the line of impact.
o Relate the relative-velocity components along the line of impact through the coefficient of
restitution.
o Apply conservation of momentum for each particle along the line perpendicular to the line
of impact.
15.5 Angular Momentum
 The angular momentum of a particle about point O, Ho, is defined as the moment of the
particle’s linear momentum about O. It is also referred to as the moment of momentum.
Scalar Formulation
 Consider a particle moving along a curve lying in the x-y plane (Fig. 15-19).
o The magnitude of Ho is
(Ho)z = (d)(mv)
(10)
where d is the moment arm or perpendicular distance from O to the line of action of mv.
o The direction of Ho is defined by the right-hand rule as shown in Fig. 15-19.
o Common units for (Ho)z are kg.m2/s or slug.ft2/s.
Vector Formulation
 If the particle is moving along a space curve (see Fig. 15-20), the angular momentum can be
determined by using the following vector cross product:
Ho = r × mv
(11)
where r is the position vector drawn from point O to the particle P. Ho is determined by
evaluating the determinant:
i
H o  rx
j
ry
mvx
k
rz
mv y
(12)
mvz
15.6 Relation between Moment of a Force and Angular Momentum
 Let Mo represent the moment about point O of a force acting on a particle, i.e. Mo = r × F.
 Using Newton’s law of motion, the following relationship can be established:
M

 H
o
o
(13)
Equation (13) states that the resultant moment about point O of all the forces acting on the
particle is equal to the time rate of change of the particle’s angular momentum about point O.
 Equation (13) also applies to a system of particles. Here, ΣMo represents the sum of the
moments of all external forces acting on the system of particles.
15.7 Angular Impulse and Momentum Principles
 The principle of angular impulse and momentum is analogous to the principle of linear
impulse and momentum. It is expressed as follows:
H O 1


t2
t1
M O dt  H O 2
(14)
where (HO)1 and (HO)2 are the initial and final angular momenta of the particle at the
instants t1 and t2, respectively. The 2nd term of the equation is the angular impulse, which
may be expressed in vector form as

t2
t1
M O dt 
 r  F  dt
t2
(15)
t1
 The principle of angular impulse and momentum for a system of particles may be written as
 H 

O 1

t2
t1
M O dt 
 H 
O 2
(16)
 In summary, the principles of impulse and momentum can be used to define the particle’s
motion. The principles are restated as
mv1 
H O 1



t2
t1
t2
t1
F dt  mv 2
M O dt  H O 2
(17)
 The vector equations in (17) can be resolved into its x, y, z components to obtain a total of
six independent scalar equations. If the particle is confined to move in the x-y plane, three
independent scalar equations may be written to express the motion, namely,
mv x 1 

t2
mv y 1 

t2
H O 1

t1
t1

t2
t1
Fx dt  mv x 2
Fy dt  mv y 2
(18)
M O dt  H O 2
Conservation of Angular Momentum
 When the angular impulses acting on a particle are all zero during the time t1 to t2, equation
(16) reduces to
(Ho)1 = (Ho)2
(19)
Equation (19) is known as the conservation of angular momentum. It states that from t1 to t2
the particle’s angular momentum remains constant.
 We can also write the conservation of angular momentum for a system of particles, namely,
Σ(Ho)1 = Σ(Ho)2
(20)