Unit - 2: Force & Motion

Module - 2.1: Linear Momentum

Contents:
Force, Momentum, Conservation of linear momentum, its applications such
as recoil of gun, numerical problems rockets ( concept only), Impulse and
impulsive force.
Momentum

• How can the effect of catching a slow, heavy

object be the same as catching a fast,
lightweight object? The answer: They have the
same momentum.

• Momentum is defined as the mass times the

velocity. The symbol for momentum is

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Momentum

• Since momentum is the product of mass and

velocity, an object's momentum changes
whenever its mass or velocity changes.
• The units of momentum are kgm/s
• is sometimes referred to as the linear
momentum to distinguish it from angular
momentum, a quantity associated with a rotating
object.
Momentum

• Momentum is a vector quantity. The momentum

vector points in the same direction as the
velocity vector.
• The following example clearly illustrates why the
vector nature of momentum must be taken into
account when determining the change in
momentum of an object.
Momentum

• The figure below shows

two objects, a beanbag
bear and a rubber ball,
each with the same
mass and same
downward speed just
before hitting the floor.
• What is the change in
momentum of each of
the objects?
Momentum

• If the beanbag has a mass of 1 kg and is moving

downward with a speed of 4 m/s just before
coming to rest on the floor, then its change in
momentum is

• A 1-kg rubber ball with a speed of 4 m/s just

before hitting the floor will bounce upward with
the same speed. Therefore, the ball's change in
momentum is
Momentum

• The total momentum of a system of objects is

the vector sum of the momentums of all the
individual objects:

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• Due to the vector nature of momentum, it is

possible for a system of several moving objects
to have a total momentum that is positive,
negative, or zero.
Impulse

• The product of a force and the time over which it

acts is defined as the impulse

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• Because impulse involves the product of force and time,

a small force acting over a long time has the same effect
as a large force acting over a short time.
• The units of impulse are the same as the units of
momentum, namely, kgm/s.
• Impulse is a vector that points in the same direction as
the force.
Impulse

• The following example illustrates how impulse is

calculated.

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Impulse

• As the figure indicates,

when a force acts on an
object, it changes the
object's momentum.
• This means there must
be a connection between
impulse and momentum
change. This connection
is revealed through the
general form of Newton's
second law:
Impulse

• Rearranging this equation, we get

• Therefore, the relationship between the impulse

and momentum change is as follows:

© 2014 Pearson Education, Inc.

Impulse

• The forces associated with impulses are often large and complex.
The figure below shows the force exerted on a baseball when struck
by a bat. The force acts for as little as a thousandth of a second,
during which time it rises to a peak and then falls to zero.

• A complex force, such as the one acting on a baseball, may be

replaced with an average force. The use of the average force, and
the time over which the force acts, facilitates problem solving.
Impulse

• Numerous
examples of
momentum-
impulse
theorem may be
seen in
everyday life.
Impulse

• A person standing under an umbrella

experiences rain, which later turns to hail. Is the
force required to hold the umbrella upright in the
hail greater than, less than, or equal to the force
required to hold it in the rain?
• The rain tends to splatter and fall off the
umbrella, while the hail tends to bounce back
upward. This means that the change in
momentum is greater for the hail. Therefore, the
impulse and force are greater in the hail.
Impulse

• The momentum-impulse theorem shows that

increasing the time over which a given impulse
acts decreases the average force. Symbolically,

• The theorem comes into play in the design of a

bicycle helmet. The materials inside a bike
safety helmet increase the time of impact,
thereby reducing the force—and the extent of
injury—to your head.
Conservation of Momentum

• The momentum of an object can't change unless

an external force acts on the object.
• Recall that the impulse is defined as follows:

• Based on this definition, if the total force

, then the initial and final momentums
must be the same, . This is momentum
conservation.
Conservation of Momentum

• The figure below shows both the internal and external forces acting
on a rider and bicycle.

• Internal forces, such as a push on the handlebars exerted by a

bicycle rider, act between objects within a system.
• External forces, such as the force the road exerts on a rear bicycle
tire, are exerted on the system by something outside the system.
Conservation of Momentum

• Only external forces can change a system's

momentum. Internal forces have no effect on a
system's momentum. Why is this so?
– Internal forces, like all forces, always occur in
action-reaction pairs.
– Because the forces in action-reaction pairs
are equal but opposite, internal forces always
sum to zero. That is,
– Because internal forces always cancel, the
total force acting on a system is equal to the
sum of the external forces acting on it:
Conservation of Momentum

• Summarizing:
– Internal forces have no effect on the total
momentum of a system.
– If the total external force acting on a system is
zero, then the system's total momentum is
conserved. That is,

• The above statements apply only to the total

momentum of the system, not to the momentum
of each individual object.
Conservation of Momentum

• Momentum conservation applies to all systems,

regardless of size.
• In a game of billiards, momentum is transferred
between the colliding balls, but the total
momentum of the interacting balls remains the
same.
• When you jump into the air, you push off the
Earth and the Earth pushes off you. The upward
momentum you gain is cancelled by the
corresponding downward momentum acquired
by the Earth.
Conservation of Momentum

• Momentum
conservation applies
to the largest
possible system—the
universe. The
exploding star in the
photo below sends
material out in
opposite directions,
thus ensuring that its
total momentum is
unchanged.
Conservation of Momentum

• Momentum conservation may cause objects to recoil.

Recoil is the backward motion caused by two objects
pushing off one another.
• Recoil occurs when a gun is fired or, as is shown in the
figure below, when a firefighter directs a stream of water
from a fire hose.
• In all cases, recoil is a result of momentum conservation.
Conservation of Momentum

• In an isolated and closed

system, the total momentum of
the system remains constant in
time.
– Isolated system: no external forces
– Closed system: no mass enters or
leaves
– The linear momentum of each
colliding body may change
– The total momentum P of the
system cannot change..
Conservation of Momentum

• Start from impulse-momentum

theorem
  
F21t  m1v1 f  m1v1i
  
F12 t  m2 v2 f  m2 v2i

 
• Since F21t   F12 t

   
• Then m1v1 f  m1v1i  (m2 v2 f  m2 v2i )

   
• So m1v1i  m2 v2i  m1v1 f  m2 v2 f
 Assignments

1. State the law of conversation of linear momentum and derive it from the
Newton's second law of motion. [CO102(N).1]
2. A bullet weighing 50 grams leaves the gun with a velocity of 30m/s. What is
the ratio of momentum of the bullet to the gun is what? [CO102(N).4]
3. Define Impulse of a force and Impulsive force. [CO102(N).1]
4. An object that has a small mass and an object that has a large mass have the
same momentum. Which object has the largest kinetic energy? [CO102(N).2]
5. An object that has a small mass and an object that has a large mass have the
same kinetic energy. Which mass has the largest momentum? [CO102(N).3]
6. A mass of 15,000 kg travels at a speed of 5.4 m/s down a track. Compute the
time required for a force of 1500 N to bring the car to rest. [CO102(N).4]
 Reference

[1] A. B. Bhattacharya & J. Pandit, “Basic Physics”,

New Book Syndicate, 2015.
[2] https://phys.libretexts.org, “Linear Momentum and
Collisions”.
[3] https://en.wikipedia.org, “Momentum”.
[4] https://www.sciencedirect.com, “Linear Momentum -
an overview”.
[5] https://www.physicsclassroom.com, “Momentum
Conservation Principle”.
 Dr. J. Pandit
Contact No. +919433195750
jyotirmoy.pandit@jisgroup.org