Conservation of Linear Momentum

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Linear momentum is expressed as the product of mass, “m” of an object and the velocity, “v” of the object. In case, if an object has high momentum, then it takes greater effort to bring it to stop.

The formula for linear momentum, p is given as:

p = mv

Here, the total momentum doesn’t get changed. This phenomenon is known as conservation of momentum. 

This article depicts more about conservation of momentum and linear momentum.

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Law of conservation of linear momentum depicts that, “when no external force is applied on the colliding bodies under a given system, then the vector summation of particular bodies of the linear momentum neither changes nor is affected by their non-to-one interaction.

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The above figure is regarding the law enabling all to know the behavior of a Newton’s cradle.

Linear Momentum Formula

Here below, we will derive the formula for linear momentum of system of particle

As mentioned above, linear momentum of the particle is:

p = mv

Further, according to Newton’s 2nd law for a single particle, it follows:

F = dp/dt ​

Where,

F = force of the particle.

For “n” number of particles, the total linear  momentum depicts as,

p = p1 + p2 +…..+pn

Individual momentum is written as:

m1 v1 + m2v2 + ………..+mnvn

Now, velocity of the centre of mass is given by:

V = Σmi​vi​​/M

mv =  Σmi​vi

If we compare the above two equations, we get,

p = mv --------(i)

Thus, it can be said that the total linear momentum of a system equals the product of velocity of the centre of mass and the total mass of the system.

After differentiation of equation (i), we get

dp/dt​ = mdv/dt​ = ma

Where,

dv/dt is acceleration of centre of mass,

ma is the force external.

Therefore,

dp/dt = Fext------(ii)

Equation (ii) shows Newton’s 2nd law to be a system of particles where the external force acting over the system is zero.

So, when Fext = 0 ---------------(iii)

Then,

dp/dt = 0

The above equation shows that p = constant.

So, when the force acting on the system is zero, then the total linear momentum of the system is either conserved or remains constant.

Here, we have proved the law of conservation of linear momentum of a system of particles.

Principle of Conservation of Linear Momentum

Let us consider equation (iii), from above, where the Fext = 0.

Here, no external force acts on the isolated system. Under that case, the rate of change of total momentum doesn’t change. This means the quantity is said to be constant.

The above explanation is the correct derivation for the principle of conservation of linear momentum.

We can say that, no matter the characteristic or property of the interaction that goes with any system, the total momentum will remain as it is. 

Now, let’s come across some examples to have a better understanding of the concept.

Linear Momentum Examples

1. Let us consider two balls that collide with each other over a billiard table. Here, the billiard table is frictionless. We can use the momentum of conservation principle to draw some conclusions and inference.

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Applying the Principle of Conservation of Linear Momentum

  • We need to consider the object that is included under the system.

  • The bodies that are in relative to the system, do identify the external and internal forces.

  • Verification of the system’s isolated position.

  • It should be ensured that the initial momentum equals to that of the final momentum.

Here, the momentum is a vector quantity.

2. Ice Skaters

Let us consider two skaters who started from rest, then pushed off against each other on ice where there is less friction. Here, the woman’s weight is 54 kg, whereas man’s weight is 88 kg. The woman moves away with a velocity of 2.5 m/sec. What is the recoil velocity of man?

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Now, given

M1vf1 + m2vf2 = 0

vf2 = - M1vf1/m2

vf2 = - (54 kg) (2.5 m/sec)/88 kg = - 1.5 m/sec

Dimensional Formula of Linear Momentum

Momentum is the measure of the product of mass & velocity. It is otherwise the quantity of motion of a body in motion.

Now, Linear momentum = mass * velocity ------(1)

The dimensional formula of mass and velocity is given by:

Mass = [M1 L0 T0] ----------(2)

Velocity = [M0 L1 T-1] -------------(3)

Substitute equation (2) and (3) in equation (1), we get,

p = mv

Or,

L = [M1 L0 T0] * [M0 L1 T-1] = [M1 L1 T-1]

So, the dimensions of linear momentum is represented by, [M1 L1 T-1]

Application for Conservation of Linear Momentum

Some of the most identified applications of conservation of linear momentum are:

  • The launching of rockets – The rocket fuel burns & pushes the exhaust gas in the downward direction. Because of this, the rocket gets pushed in an upward direction.

  • Motorboats- It pushes the water backwards & gets pushed forward so as to conserve the momentum.

FAQ (Frequently Asked Questions)

1. Find the Recoil Velocity of a 3.0 kg Pistol that Shoots a 0.060 kg Bullet with Speed of 300 m/sec. Here, the Total Momentum of the System was 0 Before Shooting the Bullet.

Ans- The momentum of a bullet and pistol is equal & opposite, as per the principle of conservation of linear momentum,

Therefore,

Momentum of pistol = - momentum of bullet

Or, masspistol * velocitypistol = - massbullet * velocitybullet

Or, 3 * velocitypistol = - 0.060 * 300

Or, velocitypistol = - 6 m/sec

2. How to Calculate the Rate of Change of Momentum?

Ans- The rate of change in momentum of an object can be identified as the mass times the change in its velocity. 

Therefore,

Δp = m⋅(Δv) = m⋅(vf−vi).

Here,

vf and vi are the final & initial velocities

Make sure to use the right signs while substituting vf & vi

For eg: A 4kg mass initially moving 5 m/s to the right rebounds off of a wall and begins travelling to the left at 3m/s.

Taking "right" to be the forward direction: we get:

vi =+5m/s, 

vf = –3m/s, and

m = 4kg.

After substitution, we get:

Δp = 4kg⋅(−3m/s−5m/s)

= −32 kg m/sec

3. What is the Significance of Linear Momentum?

Ans- The significance of conservation of linear momentum of a system or body in motion is that it retains the total momentum and is equal to the product of mass and vector velocity, provided there is an application of external force acting in it.

4. State the Unit of Conservation of Linear Momentum.

Ans- In symbolic form, linear momentum p expresses as:

p = mv

Where,

m is the mass of the system and

v is its velocity.

So, S.I unit of momentum is said to be: kg·m/sec

When we consider Newton's 2nd law of motion, the net external force is the same as the change in momentum of a system divided by the rate of change of time.