1. Momentum is defined as the product of an object's mass and velocity. It is a measure of an object's resistance to changes in its motion. 2. Angular momentum arises when an object rotates and is defined as the cross product of the position vector and linear momentum. It is conserved for objects rotating with a constant angular velocity, just as linear momentum is conserved for objects with a constant velocity. 3. Impulse is defined as the product of force and the time interval for which it acts. According to Newton's second law, impulse is equal to the change in an object's momentum. Impulse is important in collisions where large forces act over very short time intervals.

1. 1
Module # 09
Momentum, Collision & Impulse
Momentum
One of the most important quantities that we encounter frequently
in physics is linear momentum (or simply momentum). It is
denoted by P.
Consider two objects of different masses moving with the same
speed. It is common experience that it is more difficult to stop the
heavier object than lighter one. Also if two objects of same mass
are moving with different velocities then it is more difficult to stop
the faster moving object than slower one.
Thus force required to resist the motion of a moving object
depends upon two factors, mass and velocity. In other words,
there is some property relating to the motion of an object which
increases with the increase of mass as well as velocity and
decreases with the decrease in mass as well as velocity. This
property of a moving object is known as its momentum.
The momentum of an object is defined as the product of the mass
m of the object and its velocity v i.e.
P = mv

2. 2
Since, velocity is a vector quantity; so, momentum is also a vector
quantity. The direction of the momentum vector P is the same as
that of the velocity V.
In SI, the momentum has units of kgms-1
OR
NS. [N = kgms-2
& NS = kg ms-1
]
Newton's second law of motion in terms of the time rate of change
of momentum, i.e., the average force F is given by the impression
ΔP
F = ----------
Δt
and, the instantaneous force is given as
ΔP
F = Lim -------
Δt0 Δt
In case, the mass of the body remains constant during the time Δt
under the action of the force F, then
Δp = m ΔV
where, ΔV is the change in the velocity of the body in time Δt.

3. 3
Thus, we get
F = m ΔV/Δt
But ΔV/Δt has been defined as acceleration a of the body. We
can thus write the above equation as
F = ma
Angular Momentum
When a body is moving over smooth horizontal surface with
uniform velocity, then, as the velocity of the body is constant, so,
its acceleration is zero and, therefore, it is not acted upon by an
unbalanced force i.e., F = 0 and a = 0. If we define the linear
momentum (p) of a body as the product of mass (m) and velocity
(v), then, p= mv------------ (1).
The linear momentum of the object remains constant.
Now, consider an object rotating on a smooth horizontal surface
without linear motion as, for example, a spinning top. The
spinning motion of this object will continue for a long time with
uniform angular velocity as its motion is not affected from outside.
Hence, there is a definite similarity between these cases of linear
and angular momentum. In the linear motion of an object, with
uniform velocity, the linear momentum of the object is conserved;
similarly, in the case of a body rotating with uniform angular

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velocity, a physical quantity called angular momentum is
conserved.
Besides this with every nucleus and every elementary particle is
associated a certain amount of angular momentum (see
examples below).
The angular momentum L is defined by the equation
L = r x p ---------------------- (2)
Where, r is the position vector w.r.t. the axis of rotation and p is
the linear momentum.
We know that as per definition of a vector product, the angular
momentum L is perpendicular to the plane containing r and p and
its direction is determined by the right hand rule.
By inserting the value of p from Eq. (1) in Eq. (2), we get
L = r x (mv) = m (r x v) ---------- (3)
[Associative property as mass m is a scalar quantity]
From Equation (3), we can easily workout the units of L as kgm2
s-l
which can be shown alternatively equal to Js.
Magnitude of Angular Momentum
The magnitude of angular momentum L is given by

5. 5
L =  r x p  = r p sin = (r sin) p = rp
Also,
L = r p sin = r (p sin) = r p
So,
L = rp = r p
Where, r is component of r perpendicular to the direction of p
and p is component of p perpendicular to the direction of r.
Examples of Bodies having Angular Momentum
Wheels of a car rotating on their axes, earth rotating about its
axis, satellites revolving around the earth, electrons revolving
around the nucleus, etc. are the examples of bodies having
angular momentum.
Angular Momentum and Torque
In linear motion, according to Newton's second law of motion, we
have
Δp
F = --------
Δt

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Where, Δp is the change in linear momentum in time Δt. Similarly,
in rotational motion, it can be shown that
ΔL
 = --------------
Δt
Where, ΔL is the change in angular momentum during time Δt.
When torque is applied on a body, then, it causes a change in its
angular momentum.
As ΔL/Δt is equal to the rate of change of angular momentum. So
"rate of change of angular momentum is equal to applied torque".
This is statement of Newton's second law for rotational motion.
Elastic Collision
The principle of conservation of linear momentum (or simply
momentum) can be successfully applied to tackle the collision
problems. When two or more objects come sufficiently close
together so that some sort of interaction takes place between
them, which is either with or without the presence of some
external force, then, we say that a collision has taken place
between the objects.
Elastic collisions are those in which both linear momentum (or
simply momentum) and kinetic energy are conserved.

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Inelastic Collision
A collision during which the total momentum is conserved but total
K.E. before and after collision is not conserved (i.e. does not
remain constant) is called inelastic collision.
Impulse
In many cases, when the momentum of a body is reduced to zero
in a short time, then, the force causing this can have a
tremendous magnitude. When two bodies hit each other suddenly
such as a collision of two fast moving vehicles, then, the force of
impact on both the bodies is extremely large. In this context, the
concept of a physical quantity known as impulse is very useful.
Impulse is defined as the product of force and the duration of
impact.
Impulse = Force x Time
That is, the product of force and time interval for which it acts is
called impulse.
According to Newton's second law of motion, the force is defined
as the rate of change of momentum. Thus, if a force F acting on

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an object for time Δt, changes its momentum from Pi to Pf, then,
we have
Pf - Pi
F = -------------
Δt
FΔt = Pf - Pi = mVf - mVi
= m (Vf – Vi) = m ΔV = ΔP
The left hand side of this equation is defined as impulse I. Hence,
I = ΔP
Impulse is also a vector quantity and its unit is newton-second
(Ns) or kilogram -meter-second-1
(kg ms-1
).