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what is the law of conservation of momentum

What is momentum?

“Momentum of a body is the quantity of motion it possesses due to its mass and velocity.”We are aware of the fact that moving object possesses a quality by virtue of which it exerts a force on anything that tries to stop it.The faster the object is travelling ,the harder is to stop it.Similarly,if two objects move with the same velocity,then it is more difficult to stop the massive of the two.A bullet has a very small inertia due to its small mass.But why does its impact is so strong when it is fired from the gun?momentum

On the other hand,the impact of a loaded truck on a body coming its way is very large even if the truck is moving slowly.To explain such situations,we define a new physical quantity called momentum.

How to calculate momentum?

This quality of the moving body was called the quantity of the motion of the body,by newton.This term is know called linear momentum of the body and is defined by the relation:

P = m×v

In this expression v is the velocity of the mass m.Linear momentum is therefore,a vector quantity and has the direction of velocity.

Unit of momentum:

The SI unit of momentum is kgms-1 (kilogram metre per second).It can also be expressed as newton second (Ns).

Change in momentum formula:

Consider a body of mass m moving with an initial velocity v.Suppose an external force F acts upon it for time t after which velocity becomes v and acceleration is produced in the body.If Pi and Pf be the initial momentum and final momentum of the body related to initial and final velocities respectively then:

                                              Pi = mvi

 and                                       Pf =mvf

Therefore    Change in momentum = final momentum – initial momentum

or           Pf – Pi= mv – mvi

Thus the rate of change in momentum is given by:


The above equation defines force and states newton’s second law of motion as:

“When a force acts on a body,it produces an acceleration in the body and will be equal to the rate of change of momentum of the body.”This form of the second law is more general than the form F =ma ,because it can easily be extended to account for changes as the body accelerates when  its mass also changes.For example ,as a rocket accelerates ,it loses mass because its fuel is burnt and ejected to provide greater thrust.

Impulse and momentum:

Impulse is defined as:”The product of force and time for which it acts on a body.”Some times we wish to apply the concept of momentum to cases where the applied force is not constant, it acts for very short time.For example,when a bat hits a cricket ball,the force certainly varies from instant to instant during the collision.In such cases,it is more convenient to deal with the product of force and time (F × t) instead of either quantity alone.The quantity F× t is called the impulse of the force,where F can be regarded as the average force that acts during time t.

Impulse = F× t = mvf – mvi

Example:A 1500 kg car has its velocity reduced from 20 ms-1 to 15 ms-1 in 3 s. How large was the average retarding force?

Solution:By using equation of impulse:

F× t = mvf – mvi

F × 3.0 s =1500 kg × 15ms-1 – 1500 kg × 20 ms-1

Or  F = -2500 kg ms-2 = -2500 N

            =-2.5 k N

The negative sign indicates that the force is retarding one.

In case of an accident ,a person not wearing seat belt will continue moving until stopped suddenly by something before him.This something may be a windscreen,another passenger or back of the seat in front of her/him. Seat belts are useful in two ways:

  • They provide an external force to a person wearing a seat belt.
  • The additional time is required to stretching seat belts.This prolongs the stopping time for momentum to change and reduces the effect of collision.

Watch video about impulse:

 Conservation of momentum:

“The momentum of an isolated system of two or more than two interacting bodies remains constant.”Momentum of a system depends on its mass and velocity.A system is a group of bodies within certain boundaries .An isolated system is a group of interacting bodies on which no external force is acting.If no unbalanced or net force acts on as system,then according to the equation:

formula of impulse

Its momentum remains constant.Thus the momentum of an isolated system is always conserved.

Consider the example of air filled balloon as described under the third law of motion.In this case,balloon and the air inside it form a system.Before releasing the balloon,the system was at rest and hence the initial momentum of the system was zero.As soon as the balloon is set free,air escapes out of it possesses momentum.To conserve momentum ,the balloon moves in a direction opposite to that of air rushing out.

Similarly consider a system of gun and a bullet.Before firing the gun,both the gun and the bullet are at rest,so the total momentum of the system is zero.As the gun is fired,bullet shoots out of the gun and acquires momentum.To conserve momentum of the system,the gun recoils.According to the law of conservation of momentum,the total momentum of the gun and the bullet will be zero after the gun is fired.Let m be the mass of the bullet and v be the velocity on firing the gun; M be the mass of the gun and V be the velocity with which it recoils.Thus the total momentum of the gun and the bullet after the gun is fired will be:

Total momentum of gun and bullet after the gun is fired = M v + m v ….(1)

According to the law of conservation of momentum:

Total momentum of gun and bullet after the gun is fired =Total momentum of gun and bullet before the gun is fired

Therefore we get:

M V +mv = 0

or M V = -m v



Above equation gives the velocity V of the gun, negative sign indicates that velocity of the gun is opposite to the velocity of the bullet. i .e the gun recoils.Since  mass of the gun is much larger than the bullet,therefore, the recoil is much smaller then the velocity of the bullet.

Example:A bullet of mass 20 g is fired from a gun with a muzzle velocity 100 ms-1.The mass of the gun is 5 kg.Then then the recoil of the gun will be – 0.4 ms-1.The negative sign indicates that the gun recoils i.e moves in the backward direction opposite to the motion of the bullet with a velocity of 4 ms-1.

Rockets and jet engines also work on the same principle.In these machines,hot gases produced by burning of fuel rush out with large momentum.The machines gain an equal and opposite momentum.This enables them to move with very high velocities.

Conservation of linear momentum:

“When the net external force acting on a system is zero,the total vector momentum of the system remains constant.”

Suppose that the sum of the external forces acting on a system is zero.Then:

formula of linear momentum

This simple but quite general result is called the law of conservation of linear momentum.Like the law of conservation of energy,the law of conservation of linear momentum applies to a wide range of physical situations and has no known exceptions.

Conservation laws (such as those of energy and linear momentum), are of theoretical and practical importance in physics because they are simple and universal.The laws of conservation of energy and of linear momentum,frr example go beyond the limitations of classical mechanics and remain valid in both the relativistic and quantum realms.

Why does an ordinary rifle recoil (kick backward) when fired?

recoil of gun

Conservation laws all have the following form.While the system is changing their is one aspect of the system that remains unchanged.Different observers,each in a different reference frame,would all agree,if they watched the same changing system,that the conservation laws applied to the system.For the conservation of linear momentum,for example,observers in different inertial reference frames would assign different values of P to the linear momentum of the system,but each would agree (assuming ∑ Fext=0) that the value of P remained unchanged as the particle that make up the system move about.The force  F is an invariant with respect to Galilean transformations (all inertial observers agree on its measurements).If ∑ Fext=0 in any inertial frame,then all inertial observers will also find ∑ Fext=0 and will conclude that momentum is conserved.

The total momentum of a system can be changed only by external forces acting on the system.The internal forces,being equal and opposite,produce equal and opposite changes in momentum,which cancel each other.For a system of particles on which no net external force acts:

P1 +P2 +P3+……..+Pn=a constant …..(2)

The momenta of the individual particles may change,but their sum remains constant if their is no external force.Momentum is a vector quantity.Equation 2 is therefore equivalent to three scalars equations,one for each coordinate direction.Hence the conservation of linear momentum supplies us with three conditions on the motion of a system to which it applies.The conservation of energy,on the other hand supplies us with only one condition on the motion of a system to which it applies,because energy is a scalar.

If ous system of particles consists of only a single particle, then equation( 2 ) reduces to a statement that when no net force acts on it the momentum of the particle is an constant,which (for a single particle) is  equivalent to stating that its velocity is constant.This is simply a restatement of newton’s first law.

Momentum and explosive forces

There are many examples where momentum changes are produced by explosive forces with in an isolated system. For example, when a shall explodes in mid air, its fragments fly off in different directions. the total momentum off all its fragments equals the initial momentum of the shell. suppose a  falling bomb explodes into two pieces. the momenta of the bomb fragments combine by vector addition equal to the original momentum of the falling bomb.

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